It's not that someone would say others were cheap, or that people are moochers (why would you keep going out with those people?!). It'd be odd and seems unnecessary and just strange.
If you do that, then you do. I never have, I've never been with anyone who has, so why would I? Also, I think it'd be strange to ask.
I mean I can't really think where I even would. At a bar? Either people serially buy rounds, or people go to the bar itself, order and pay for a drink when they want one, or someone is running a tab, or it's a thing involving waitstaff in which case I can't imagine the pain in the rear of them trying to keep up with separate checks.
At a neighbourhoody restaurant? How hard is it to simply look at the bill? I'd think they'd think I was strange for asking; I've never seen it done, though it's not like I pay attention to how other tables are paying but you'd think I'd have run into anyone doing at some point it if it were common here.
At a nice place? As Janepod said, that's just, again, weird. Usually someone pays and other people just pay them if it's being split not traded off.
If separate checks are your norm then go nuts but some of us have managed fine without and if it ain't broke...
As to the 'drama' it usually works out fine. On occasion (usually with a big group at a sports or other bar) it'll be a bit short because someone can't add or forgot something was theirs, or it'll be over because everyone overestimated. No big. People toss in extra or take back some or the server gets a really nice tip (once during playoffs our fave sports bar waitress ended up with a 50% tip). No big.
Separate checks also seems like it'd take longer, with everyone having to pay the waitstaff as what if a whole bunch of people used cards...