Yes, both are said.
But I'd argue that we say "are we meeting at the restaurant" (beach, house, club, etc), leaving out the implied "up", more often.
We know it usually means "meet up" as opposed to "meeting for the first time". It's always obvious which connotation the phrase means, by definition of we've need before or not.
I have another issue with this sentence: we say: "meeting AT the beach",
"meeting On the beach" !
If you wanted to use nosotros, that would be "Nosotros nos encontramos" - in this answer, nosotros is omitted because it's understood from the verb. "Nos" is needed either way, because it's the object of "encontrarse" here. The sentence translated exactly would be more "we found ourselves on the beach" but the way it's used is to meet up.
Encontrar does mean "to find", but when it's used reflexively (encontrarse as opposed to just encontrar) then it also means "to meet". There's a good reference about the use of encontrar[se] here: https://www.realfastspanish.com/vocabulary/encontrar-vs-encontrarse
Note that the "to meet" of encontrarse is a little different than the "to meet" of conocer. Conocer means "to meet" as in to meet for the first time, to be introduced to someone. Whereas encontrarse means "to meet" as in meeting up with someone you already know, like arranging to meet your friends somewhere to hang out. So "nos conocemos en la playa" is also a valid sentence, but with a slightly different meaning from "nos encontramos en la playa".