I have two Trello accounts and have never had a problem switching between them. I have one for work and another for personal stuff, just like you suggest, and it works fantastic. If I'm logged in to one and want to check the other I just log out, enter the credentials for the other account and it works exactly like I would expect. There's no mixing at all, they're completely distinct, which is exactly how I want it, and exactly how it should be, IMO.
Google, on the other hand, is an absolute mess. At this point I've resigned myself to the fact that my work Gmail/Google+/Google/YouTube account and my personal Gmail are inextricably linked together and there's no way I can unlink them. It's such a disaster that it was a big factor in my decision to start moving my personal accounts away from Google.