Never mind the stupid EU referendum, there were some truly regrettable things about the Queen's Speech: chiefly, the proposals to tackle benefit and healthcare "tourism" (modern Conservatives use this previously bland word, used for simply exploring foreign countries, as a euphemism for deliberate exploitation. Who knows where tourism tourism will stand, once this mucky debate is over?)

There are classic arguments against the whole premise. EU and non-EU migrants combined cost the NHS only £7m; their cost is far smaller than their number, proportionally, because they are younger, on average, than indigenous Britons; they are less likely to be ill and they are less likely to have started a family.

But a subtlety has been missed, which is the assumption that EU migrants, specifically Polish people (as the largest single group) are coming from a third-rate health service of their own to a first-rate one of ours. This is, apparently, not the case; indeed, many Polish immigrants will go to some lengths to avoid NHS "tourism", up to and including paying for their care.

Since 2007, private Polish medical centres have sprung up all over the UK; there are at least 20 in London, they exist in Manchester, Reading, Bristol and Glasgow. Nobody collects a database of them, but wherever there's a Polish community, there's a private clinic.

Ewa Rybol, 27, is the practice manager at Tooting Medical Centre. "We get Canary Wharf Polish here. They're developing their career here, they probably work in a bank. But we also have people who do simple jobs, waitressing, building, cleaning." The dermatologist at the practice (who wished to remain anonymous) agreed: "90% of the people I see don't make a lot of money, but they are still willing to pay, even when they speak good English."

The language barrier is usually raised first, when people are asked why they'd go private, but I think that's out of politeness. Certainly Alicja, 25, who paid £60 to see a dermatologist in Tooting for her eczema, sounded less than plausible when she explained: "I wasn't sure that I would be able to understand a GP. Medical language can be quite specific." (her English was perfect).

The dermatologist elaborated: "I wouldn't want to make a generalisation, but where language isn't a problem, there is sometimes disappointment. GPs don't refer people to specialists. There must be a reason for that, and I guess it's economical."

Piotr Miklewski, a 29-year-old practice manager from PMC in Ealing, said: "There is this stereotype that the UK prescribes paracetamol for everything."

Wiki, 29, who broke her rib ice-skating, said: "And they don't x-ray you. In Poland, you would always be x-rayed for a broken rib in case it was endangering your lung". She added, "The doctor told me to take a paracetamol".

Miklewski said the differences between the two health systems, Poland's and the UK's, were most noticeable when it came to pregnancy. "In the UK, you don't do the ultrasound scans until a certain age of the pregnancy, which we do almost from the very beginning.

This is one of the things that mainly drives people to go to see our gynaecologist. They can expect to have a scan from the very first moment that it is required."

Theresa Drzewiecka, 55, is a nurse at the Tooting practice, and said that people also saw her because they were confident of her training. "They think maybe if you go and get a blood test from an inexperienced nurse, it will hurt. If you see a Polish nurse, you know it won't hurt" (in Poland, they train for five years and then have three to four years' apprenticeship; this is somewhat longer than the three-year degree course it takes in the UK).

Many people bring up waiting times, both for a doctor's appointment – "you can wait a week for an appointment," Alicja said, "you would never wait more than a day in Poland".

And there is much more acquiescence in Poland to the idea of paying for some things but not others.

You can see this from the clinics themselves. In Tooting, it is certainly sprucer than the local GP's surgery – more like a beautician than a doctor – but it doesn't look like Harley Street and there is none of the thick-carpeted hush that bankrupts you. "We have really nice prices," the receptionist, 28-year-old Marta Baczewska says. "English customers come in here as well, when they hear about us."

Personally speaking, I would never knock the NHS for its paracetamol name, the reputation it has for never referring anybody, never prescribing antibiotics, never scanning anyone and trying not to waste x-rays. It makes us sound thrifty and stoic.

But if we think people are travelling here to make the most of our health service, we're dreaming.