I’m pretty sure that amoebas reproduce by strict mitosis (making two copies of their DNA and splitting into two daughter cells, all of which are genetically identical (assuming no transcription errors)), so no, they don’t exchange genetic material with other amoeba.
But I imagine an amoeba deciding to split would go something like this:
“Hey, self, baby, you’re some pretty hot stuff. I like you, or, well, me, a whole lot. I think I should get it on with you, self.
“Oh yeah, oh, awesome, lets use some DNA helicase and unzip those genes.
“This is Nasty! I like it! Oh, man, we’re splitting our nucleotides something fierce!
“Oh, looks like that’s over with. Here, let me take half the covers.
“Oh, hey, we are two different people now. How sad the universe is for splitting us apart.
“Or was it by our own volition?”
Yeah, it’d be something like that.