Was sind die Unterschiede zwischen den Wörtern:
Wiktionary nach sie sind Synonyme. Sind die Unterschiede regional? Ist ein Wort formaler als die anderen?
They all mean together, somehow, but you can't all use them in the same sense. I'll try to give you a few examples:
Gemeinsam means together, united, as a community:
Beisammen means assembled, collected:
Zusammen mit means together with (zusammen kommen = come together):
Miteinander means conjointly, together, side by side, literally "einer mit dem andern" (one with the other) and highlights the relationship and cooperation between people. It's also the opposite of gegeneinander:
Differences are subtle and not easy to explain, I am afraid. You'll probably want to remember a few fixed collocations to get you started.
The difference is mainly function.
This is a pure adverb. That means the sentence
is wrong. It means "together" in sense of "together we do"
This is an adverb too so again the sentence
is wrong. Other than "gemeinsam" it implies more interaction between the parties.
And here's an example in which "miteinander" wouldn't work:
It sounds odd, as eating doesn't allow for the degree of interaction. (We ate with each other... what?)
This is (mostly) an adjective so it does work in the sentence we had, but it doesn't work in combination with a verb.
As the other answer shows, it is possible to use "beisammen" in combination with verbs like "sitzen". It kind of feels like a local-adverb then though. Of all the words, this is the least common one I would say.
This can be used as either, adjective or adverb.
It can be a synonym for "beisammen" and "gemeinsam" but not really for "miteinander" as "zusammen" lacks reciprocity.