Now when watching Londyńczycy there is this case of a girl who have to send money back to Poland, so that her sister can pay for an operation for her baby.
I don't know the show that you're talking about, but it's likely that this is referring (indirectly) to the most shameful thing in Poland - the fact that doctors are still taking bribes, particularly where operations are concerned. If it's not that, then it'll be an operation that's not offered under the Polish NFZ (national healthcare scheme). This isn't entirely uncommon - you might find that parents want a certain treatment that's available in America for $100k - and the Polish system just doesn't have the money to pay for that kind of thing.
Now I'm wondering if this is common in Poland? Are people in Poland enforced by the government to have a health insurance? And does such an insurance covers all needed life-saving operations?
If it's bribery, then it's still sadly common.
The system in Poland is absolutely weird in terms of health insurance. Instead of having universal cover like in the UK, with taxes paying for everyone - instead, there's a bureaucratic mess involving people needing to be insured. It's compulsory for most people (as you can't avoid the payment to the state insurance company), except many people don't have to pay for whatever reason (spouse works and is covered through him/her, students/children are under their parents insurance, they're retired/disabled, and I'm sure there's more). The even more crazy part is that everyone employed has to get a little booklet stamped MONTHLY by their employer to prove that they're entitled to the insurance - which is just ridiculous.
Some people can fall through the net though - for instance, if you aren't working because your parents are paying you to bum around at home, you won't have health insurance. This is where the whole system is clearly mad - such people are forced to register as unemployed in order to be covered, even if they can't/won't want any benefit payments.
But the state health insurance is completely hit or miss. Some aspects are fantastic, such as the way that many private specialists also operate under the NFZ. But some hospitals are absolutely dire - but this is balanced by some being well ran and decent.
As for the operations - there are plenty of horror stories where the NFZ system has refused to do operations (only for the operation to be done after the payment of a bribe). Emergency life saving operations will of course be done - but if someone has cancer and needs an operation to prolong (but not cure) their life, then we're back to the issue of bribery.
In general I don't think that a hospital would not operate a baby because his/her parents are not properly insured.
Again, it's the issue of bribery - while the NFZ might be happy to pay for the operation, you might find that the operation keeps getting delayed until the doctor has his hands filled with silver.
Things will change, but as long as ZUS (State insurance company, dealing with pensions and health care) has to subsidise people who didn't pay very much in real terms (ie, when the Zloty was freely convertable) into the system - then the NFZ will remain dreadfully underfunded. There are people out there, retiring today, who have paid only 20 years of contributions and will be expected to live for 30-odd years on free healthcare and a pension. And people wonder why their ZUS contributions are so high...