I guess you can do the routine stuff in private health care, where they actually care about the quality of service (so they should be able to communicate in English). It's not really expensive too; in Warsaw appointment with a specialist (cardiologist etc.) is $25-$50.
For serious stuff (like surgery) it's best to go with public health care though.
I dunno, I think it's very dependent on the health provider in question. I've had some shockers from Lux Med, both in terms of incredibly bad service (eg doctors clocking off early and then claiming I turned up late, despite being obviously and easily provably false), and issues with a lack of English speakers, despite being assured that it wouldn't be a problem.
In fact, they're bad enough that I actively campaign against them.
Whoops, turned into a rant there by accident, sorry about that :)