I cannot speak to meeting, but I can speak to the fact that in written interactions, a monarch addresses fellow monarchs as "sister" or brother", due to a concept that all monarchs are members of the same family, so a letter would be addressed to, "Sir My Brother" or "Madam My Sister".

If they are actually related, then that is also specified, as in "Sir My Brother and Cousin" or "Madam My Sister and Niece", which makes the already unwieldy Hapsburg, Hanover and Bonaparte family trees seem even worse than they are.

For sources, you can check Elizabeth Stuart, Queen of Bohemia's letters, the letters of Napoleon, Debrett's collection of state papers, and so on and so forth.

I can only talk about Napoleon's interactions with European monarchs, but I'd say other such regal meetings would go down in a similar way.

His first meeting with Francis II, Emperor of Austria and the Holy Roman Empire, took place in late 1805, just days after Napoleon's victory at the Battle of Austerlitz. The meeting is reported to have been cordial and very productive - the two managed to come to terms of peace in 90 minutes.

Napoleon first met the Russian Tsar Alexander in 1807, in a meeting that would result in the Treaty of Tilsit - an end to the War of the Fourth Coalition and Russia forming an alliance with France. Napoleon sought to establish a warm personal relationship as well as an effective working one. They concluded the peace treaty the very same day, but spent the next couple of days reviewing each other's guardsmen, exchanging decorations, talking discussing the future of the Ottoman Empire, and most importantly, drinking together. It was a quite cordial relationship, with Napoleon even going so far as to consider Alexander his friend.

Basically, it all came down to mutual respect. The three Emperors believed themselves to be on equal footing as the most powerful sovereigns in Europe, so even though calling their relationship friendly is a bit of a stretch, they were, by all accounts, quite cordial and respectful to each other.

That would be a rough overview about Napoleon's relationship with other sovereigns, but I'd be glad to answer any follow-ups.

Would two monarchs ever really meet? Would they not have people for that, like envoys?