I can't identify the varieties specifically, but they are beautifully grown rhizomatous begonias. Probably have names. They don't need much sun, as yours demonstrate, tho' they'll put up with a little. Dappled shade is about all. Rhizomatous begonias will use their rhizomes to walk right out of their pots--you can re-pot periodically, placing the back end of the rhizome up tight against the edge of the pot and getting the growing tip in the middle, or you can just remove the front part and pot it separately. It'll make the back send out new shoots to make a more 'balanced' plant, and you can pot up the front for a new start. There are other methods for keeping them 'young' and growing vigorously too. Maybe others will plug their methods!

It looks a lot like 'Bethlehem Star', but I cannot be positive from the pic. Different conditions also grow begonias differently. Greg in HOT DRY Tampa Bay

Thank you, greenelbows1, for the helpful growing tips, now I know better what to do with them. I saved a copy of your instructions and hopefully I can get some nice looking and substantial plants out of my little rootlings.

Thank you, gregsytch, I think you are correct, I googled some pictures of Begonia 'Bethlehem Star', and that appears to be what it is, you have amazing ID skills to pick that out from my terrible pictures!