. . . at least if your church is a Presbyterian church.
In the presbyterian system of church government, ecclesiastical authority is collegial. Elders do not function alone, but as parts of larger bodies of elders. Individual elders function as parts of sessions, in submission to each other. Sessions function under larger bodies of elders called presbyteries which have authority over the sessions and elders that make them up. Presbyteries are subject to the authority of larger binding synods made up of elders from the various presbyteries, and these synods can be provincial, national, ecumenical, etc. As American Presbyterian theologian Charles Hodge summed it up, "The Presbyterian
doctrine on this subject is, that the Church is one in such a sense that
a smaller part is subject to a larger, and the larger to the whole."
In light of this, here is a question you can ask to find out if your church (assuming it is presbyterian) accepts the objective legitimacy and authority of another church: If the other church called on your church to join them in a common council to discuss and try to resolve differences in doctrine and practice, and this council would be binding on both churches, would your church accept the call? If the answer to this question is yes, then your church recognizes the legitimacy and authority of the other church. If the answer is no, then your church does not recognize the legitimacy of the other church. Since church power is inherently collegial, a refusal to submit to the authority of the other church by joining with them in a mutually-binding council amounts to a declaration that that church does not possess ecclesiastical authority.
To give a concrete example, take the OPC and the FPCS. If the FPCS called the OPC to join them in a mutually-binding council to discuss and resolve doctrinal and practical differences, would the OPC accept the call? The answer is no, they would not, because they would not want to grant the FPCS power over the doctrine and practice of the OPC. Since the OPC professes a presbyterian view of church government, this means that the OPC does not recognize the objective legitimacy and authority of the FPCS. The FPCS, likewise, would not accept a call from the OPC to join in a mutually-binding council, so the FPCS does not accept the authority of the OPC either.
For more, see here and here.
UPDATE 2/17/15: Some may try to escape the force of the above
reasoning by suggesting that their denomination (D-1) is indeed in
mutual submission to another denomination (D-2) because D-1 is perfectly
willing theoretically to join with D-2 in a mutually-binding council,
even though D-1 refuses to recognize any way in which such a council
could legitimately be called, and so there no practical possibility for
such a council ever to happen. So, theoretically (although I do not
believe this claim is actually made), the OPC might claim to be in
mutual submission to the FPCS because it is perfectly willing to join
with the FPCS in a mutually-binding council--although no such council
could ever actually happen because the OPC refuses to recognize any
acceptable way to call it. In this way, theoretically, denominations
that want to think of themselves as consistently presbyterian but which
don't want to acknowledge that they reject the authority of other
denominations might try to evade the presbyterian logic.
this evasion won't work, because to acknowledge something in theory
while completely rejecting it in practice amounts to the same as
rejecting it in theory (except that it adds hypocrisy to the mix). What
would we think of a church which claimed to be presbyterian but which
recognized no way to have presbytery or higher synodical meetings,
making it impossible ever to have them? This would be presbyterianism
on paper but not in reality. Such a church would rightly be regarded as
in actuality a congregationalist church. Or what would we think of a
church that claimed to reject icons in its creed but which actually
tolerated them without any concern at all and refused to hear any
discipline case against their use? We would say, rightly, that such a
church is not really opposed to icons. It must be recognized
that a church's practice will not always perfectly match its profession
(just as it is with each of our individual lives as well), and there
must be some toleration for flaws in practice, but there is also a limit
to such toleration. A church which completely and fully in practice
rejects what it claims to affirm in theory is just playing games and its profession should not be taken seriously.
that is why this evasion does not work. If D-1 wants its claim that it
is in mutual submission with D-2 to be believable, it must make
reasonable efforts to practice such mutual submission. At least this
must translate into recognizing some possible, practicable
way in which a mutually-binding council could be called and actually
allowing such a thing to occur if it is called. So if the OPC wishes to
claim (and I do not say that it does) that it recognizes the
presbyterial authority of the FPCS, it must show that it is willing to
submit to the FPCS's authority by acknowledging (and following through
with if called upon to do so) some practical way in which a
mutually-binding council between these denominations could be called.
Without such a tangible expression of submission, a profession of
such submission would be nothing more than mere pretense.