Most often, S2S contacts occur either by arrangement with a known other activator, or unexpectedly when a happy coincidence of your activations occurs.
The arranged S2S contacts don't need any special calls or procedures. It is good practice to confirm your summit codes so that both operators are in no doubt that the contact is S2s.
The unexpected contacts usually occur when an activator has called CQ or QRZ, and one of the callers will either give his/her callsign and add "Summit to Summit" or "S to S" or in CW will add "S2S", or will possibly just say "Summit to Summit" as traditionally priority is given to such contacts.
Unless it is one of those special days like the 1st of January, when by tradition in Australia, for example, there is an "S2S-fest", there is usually no point in calling CQ for S2S contacts. But that all depends on the level of activity and whether you know there are likely to be other activators on summits at the same time as you are. Calling CQ S2S when no other activators are visible on Sotawatch would be optimistic.
In the SOTA contacts database, you need to record the contact as an S2S contact in your chaser log. That's where S2S contacts are recorded. Both summit codes need to be recorded on S2S contacts, otherwise you won't get S2S credits. Note that the contacts are also recorded in your activator log. The activator entry adds to your activator score, the S2S/chaser entry adds to your chaser score and your S2S score.
Hope this helps.