Like the Sheriff and Wilbur who were advised they were still under oath.
Hey! You're not old enough to drink! Now go and die for your country!!!

You specifically asked about
after
the witness testifies.
That's not so clear-cut.
Yes, a witness will leave the stand. And he will be "finished" testifying.
However, we don't know if he might be called back again later, for some reason.
So, it's not always that clear-cut and "black and white" that the witness is in fact "finished" testifying.

Generally speaking, anyone in the public including a witness would be allowed to sit in on the trial.
Now, one side or the other can request that the judge prevent that.
Sometimes that is done so that one witness (let's say Witness #2) cannot get the benefit of hearing the testimony of another witness (let's say Witness #1).
So, one of the lawyers and the judge will not want Witness #2 to hear what Witness #1 has to say. The fear being that Witness #2 will change his story to make sure that it "matches" what Witness #1 said.