According to Beth Din of America, Reuvain can end up with Shimon in Beit Din in three ways:
Parties may end up before a beth din either:
under a preexisting contractual provision that requires the resolution of a dispute before that beth din,
after they have mutually agreed to have that beth din decide a dispute
because they have chosen that beth din through the hazmana (summons) process.
There is more informaton here http://www.bethdin.org/cases.asp.
If Shimon does not want to go and Reuvain summons him, he is only HALACHICALLY required to go, but not legally (civil law). There are a few options (http://www.bethdin.org/docs/PDF1-Layman%27s_Guide.pdf):
In the event that there is an existing agreement to come to a particular beth din (for
example, if the parties signed a contract which refers any disputes to a particular beth din),
the beth din can hold a din torah in the absence of the person who was summoned. The
beth din considers the evidence that the claimant brings and issues a ruling on that basis.
Otherwise, the summoning bet din generally issues a “heter arkaos,” which grants
permission to the plaintiff to go to secular court, so the defendant does not simply avoid a
A summoning beth din may issue a “seruv,” or a contempt order. A seruv is simply a public
declaration by a beth din that someone was summoned to beth din but refused to meet their
obligation under Jewish law to appear in front of the beth din. Sometimes, Jewish
communities or synagogues impose sanctions on such people, such as not giving them
aliyos (being called up to the reading of the Torah) or refraining from social interaction, to
pressure the person to meet their obligation. The beth din is not necessarily involved at
that point – the community or synagogue decides what measures are appropriate.
Occasionally a summoning beth din may issue an “ikul,” a restraining order. The Beth Din of
America seldom does so.
At least at the Beth Din of America, the decisions are legally binding because the Beit Din acts as a legal arbitrator. http://www.bethdin.org/arbitration-mediation.asp
Based on the below paragraph (http://www.bethdin.org/cases.asp) it would seem that for these purposes a Beit Din really is just 3 arbitrary judges (knowlegable, observant men), and they may be chosen in any way as long as the parties agree, or a special procedure is followed.
Someone receiving a summons is generally not obligated to agree to go specifically to the summoning beth din, although he or she is obligated to either settle the case or go to a beth din. If the person being summoned does not want to go to the summoning beth din, then he or she must propose an alternative beth din. If the parties cannot find a mutually acceptable beth din, a “joint beth din” is formed by a procedure called “zebla” or “zabla.” In zabla, each side picks one judge. The two judges that were picked select a third judge together, and the three judges together form the beth din that will decide the case.