For more hygienic flossing with stick flossers, run a trickle of water while flossing. Rinse the floss stick after each tooth. This prevents plaque being transferred from tooth to tooth.
Who's going to go that trouble? These are a convenience item, portable and handy. I use these while driving, in bed at night, etc., NOT while standing over a sink.
If there's plaque on one tooth, it's going to get to the other teeth anyway.
I like to floss in the shower. I usually do it while I'm conditioning my hair. Before, I would avoid doing it because it was so messy. The shower just washes all the yuck away.