For the second part of the question, might I suggest looking over our FAQ on pirate history in the r/Askhistorians wiki. In particular one of the questions that starts with "How likely was it to encounter pirates?" and another that starts out with "I'm a typical sailor in the Caribbean..." It will answer most of your questions. But, to address the first part more specifically, two vessels could plan to meet at a particular landmark on nearby land - but a planned meeting at sea would be hard to arrange, I don't really recollect anyone doing such a thing (since, without land markers you would have to use a combination of measuring the latitude and dead reckoning (which isn't that accurate) to do so). Beyond meeting at planned landmarks on the coast (or ports), all your other meetings are by chance, though, as you point out, the chances of meeting another vessel are raised significantly when you sail in one of the trade winds that dominate the Atlantic World. The method did work, since a number of pirates did end up encountering vessels that way. Also knowing familiar landmarks where ships would regularly sail by could help. While it's very possible to encounter practically no ships on a voyage, it's also possible to encounter many.

For an intentional, planned meeting , usually a series of rendezvous would be established since nothing at sea in that period could ever be remotely certain and since the ships were of course at the mercy of the wind.

So for example ship 1 and ship 2 might both plan on cruising between a given latitude/longitude during the last week in May

If they don't find each other during that week they might both plan on cruising between a second given latitude/longitude or maybe off some particular landmark

There could be a lot of these rendezvous or they could also plan on meeting in a certain port where both intended to refit.

But ships all had water, food and other supplies in limited capacity so hanging around the ocean for too long was never an option, especially since one ship would never really know whether or not the other had been wrecked.

As for spontaneous meetings, the mainmast of a large ship rises very high, and a lookout with good eyes can get a very wide view of the ocean from the top. Just as Whalers would use the famous crow's nest at the top of their mainmast to spot whales.

Plus many shipping lanes and fishing grounds were frequented by the multiple ships (Think about the straight of Gibraltar, the only way into the Mediterranean).

So in a voyage from Malta to Gibraltar for example, you would probably see a bunch of merchant vessels and navy vessels. For a longer voyage in a larger ocean, like going from England to India? Yeah you might go months without seeing a ship during some parts of the voyage.

As for ships that were sighted, it would depend on what kind of ship you were (Navy vessel, merchantmen, whaler, ect) and what you supposed the ship you sighted to be.

For example, in 1807, when the French and British were at war, a British Whaler might shy away from what they thought might be a French Naval Vessel which might capture them.

If it were, to take a different example, a British Navy Vessel coming up on another British Navy Vessel? Then yeah they would probably communicate, signaling each other first with a gun (pretty easy to hear at a long distance in the open ocean) and then through signal flags hoisted to the tops of the masts.

If two ships really wanted to speak, or if a ship was for example required to show papers to a cruising navy ship, than ships all had smaller boats they carried that could take the captains or officers back and forth.