This form of church government has implications for the meaning of denominational separation. When churches are divided from each other in distinct denominations, there is no formal, binding, mutual accountability between the churches. Since the authority of elders is inherently collegial and requires in its essence mutual submission between elders, the refusal of two or more denominations to acknowledge mutual submission to each other necessarily (assuming a consistent presbyterian practice) implies that the denominations do not grant to each other legitimate presbyterial authority. That is, they do not acknowledge each other to possess such authority. Such refusal to acknowledge authority might be just or unjust. If Denomination A possesses legal authority as a church and rightfully refuses to acknowledge such authority in Denomination B, then it must be regarded as an objective fact that Denomination B does not possess legal authority. This is because God has promised to ratify the decisions of church courts when they are just and consonant with his Word. However, if Denomination A is acting unjustly in its refusal to recognize Denomination B's authority, this act is not ratified by God and is in fact a schismatic act.
This separation, however, does not imply that the divided denominations do not regard each other as members of the visible church in a de facto sense. God is merciful, and we have reason to hope with a judgment of charity that the presence of his Body is manifest in the world far beyond the confines of the legal body of the church. God has his people and manifests his saving work in many denominations, and the visible Body of Christ can thus be said to be present within these denominations. Christians in these denominations possess spiritual gifts, including gifts of preaching, administration, and oversight. God frequently works through these gifts to providentially guide and feed his church in the world, even though the gifts ought to be exercised with objective legality in communion with the legal church. Just as God, in mercy, works through the ministry of baptists, and even blesses the children of baptists despite their failure to give to their children the required sign of the covenant, so we have reason to think also that he works in many ways providentially through the de facto Body of Christ even where objective de jure legality is absent.
It is our conviction that the Free Presbyterian Church of Scotland is justly separate from other denominations, but not the other way around, and so the FPCS has de jure authority and legitimacy while other denominations lack it.
For more, see here.
UPDATE 10/31/14: I've always liked the statement I first wrote up explaining my views on these subjects, way back in May of 2012 when I was still a member and a ruling elder in the OPC, so I though I'd paste a bit of it here:
Churches obviously claim authority.
Church officers claim authority to fulfill the various functions of
their offices; sessions, presbyteries, and broader assemblies claim
authority to function as legitimate synods; etc. Whenever there are
multiple denominations, each denomination, by virtue of its existence
separate from other denominations, is making a two-pronged statement:
“We possess legitimate authority, and other denominations don’t.”
We can think of the church de facto and de jure.
Considered de facto, the church of Christ is of course broader
than any particular denomination. That is to say, the Body of Christ
and its essential activities and functions can be found in many
different places and denominations. Considered de jure,
however, each denomination holds itself alone to be the visible
church of Christ. By existing as a separate denomination, the OPC,
for example, recognizes the authority of the OPC General Assembly, of
the individual presbyteries of the OPC, and the sessions that make up
those presbyteries; but it does not recognize the authority of the
assemblies, presbyteries, and sessions of other denominations. The
OPC does not invite to its General Assembly meetings, as fully
recognized voting members, ministers from other denominations. They
are treated as being without authority. If the Free Church of
Scotland comes to a decision in its General Assembly and mandates
that it be enforced in presbyteries and sessions, the OPC treats this
decision as being without authority. There may be a de facto
recognition that the Body of Christ, the visible church, is present
in these other denominations, but there is no de jure
recognition of the authority of these other bodies. The full
recognition of the authority of the authoritative bodies of other
denominations would amount to full union with them so that there
would no longer be two distinct denominations.
Now, this fact, I have come to see, is important in terms of selecting the denomination one is going to be a part of. As there are a multitude of conflicting Christian denominations, they cannot all be right in their claims to authority, for they all exclude each other. This means that there is an objective question to answer: Which denomination, if any, is the legitimate heir to the de jure Church of Christ? There have been many denominational splits in the history of Christianity. It is theoretically possible to have a split in which both sides are equally wrong. It is not possible to have a situation in which both sides are fully justified in the split, for otherwise there would be agreement and no split. A denominational split always involves sin and/or error. There is always heresy and/or (non-doctrinal) schism. Most of the time, there will likely be one side that is right and one side that is wrong, or at least one side that is more right than the other. When a split occurs, it is necessary as far as possible to ascertain which side is right and to follow that side, for that side will possess the continuing authority of the de jure church while the other side will not. (Again, I am not saying that both sides can’t be de facto true churches. I am saying that they cannot both be de jure true churches, because in their mutual separation they exclude each other. The side that is right, then, being legitimate in its exercise of authority involved in the split, will retain that authority, while the side that is wrong will lose it, as it has had its authority rightfully revoked by the other side.) The question of which denomination is the rightful heir of the true de jure church is thus an objective question that we all must answer, and there will be an objective answer to that question that will not differ depending on one’s location, culture, language, nationality, etc. (except insofar as one’s conditions prevent one by some means or another from knowing about or being unified with what otherwise ought to be considered the rightful heir). We must first look at the churches currently existing and determine which is the most orthodox, and then we must factor in historical information regarding how they came to be separated from each other, and from these we can determine which denomination ought to be regarded as having rightful de jure authority. We would then have a prima facie moral obligation to join that denomination and not to support other denominations (by doing anything that would attribute de jure authority to them, by joining them, being an officer in them, etc.).
Now, this fact, I have come to see, is important in terms of selecting the denomination one is going to be a part of. As there are a multitude of conflicting Christian denominations, they cannot all be right in their claims to authority, for they all exclude each other. This means that there is an objective question to answer: Which denomination, if any, is the legitimate heir to the de jure Church of Christ? There have been many denominational splits in the history of Christianity. It is theoretically possible to have a split in which both sides are equally wrong. It is not possible to have a situation in which both sides are fully justified in the split, for otherwise there would be agreement and no split. A denominational split always involves sin and/or error. There is always heresy and/or (non-doctrinal) schism. Most of the time, there will likely be one side that is right and one side that is wrong, or at least one side that is more right than the other. When a split occurs, it is necessary as far as possible to ascertain which side is right and to follow that side, for that side will possess the continuing authority of the de jure church while the other side will not. (Again, I am not saying that both sides can’t be de facto true churches. I am saying that they cannot both be de jure true churches, because in their mutual separation they exclude each other. The side that is right, then, being legitimate in its exercise of authority involved in the split, will retain that authority, while the side that is wrong will lose it, as it has had its authority rightfully revoked by the other side.) The question of which denomination is the rightful heir of the true de jure church is thus an objective question that we all must answer, and there will be an objective answer to that question that will not differ depending on one’s location, culture, language, nationality, etc. (except insofar as one’s conditions prevent one by some means or another from knowing about or being unified with what otherwise ought to be considered the rightful heir). We must first look at the churches currently existing and determine which is the most orthodox, and then we must factor in historical information regarding how they came to be separated from each other, and from these we can determine which denomination ought to be regarded as having rightful de jure authority. We would then have a prima facie moral obligation to join that denomination and not to support other denominations (by doing anything that would attribute de jure authority to them, by joining them, being an officer in them, etc.).