No one except a vet should be putting any animal to sleep.
But to respond to your comment, we do. Animals that are un-rehomable due to illness/aggression we humanely euthanise at our shelter. We even 'trade' with the city pounds that we will organise to have these animals PTS if you allow us to bring our vet in for a day to neuter the dogs the pound. We don't want to rehome dogs to the UK, we actually lose money by doing so - it costs around £550 per dog to bring it (legally and safely) to the UK. We charge a £300 adoption fee. For every dog we rehome in the UK we home another 3 in the country we operate.
But - I cannot stress this enough - pounds there aren't like here. Some city pounds don't even have workers there from Fri eve-sunday morning, so the dogs are not fed/cleaned in that time. One pound (that we have since taken over) used to put small dogs in with big dogs on a Friday night so the big dogs would kill and eat the small ones. One vet (now in jail) was stitching up bitches after neutering with ribbon "for fun". These dogs were dying in agony.
I cannot describe the levels of depravity SOME (not all) Eastern European shelters succumb to. During the floods a few years ago we went to a flooding shelter and BEGGED them to release all their 36 dogs to us. They refused, left all the dogs chained in cages. They all drowned. To them this was a gift - their shelter was empty. They made their money (they get paid per dog that leaves the shelter whether dead or alive) by getting rid of all the dogs AND they got to laugh at us for caring about 'stupid dogs'.
And, of course, there is legal resistance within the countries. We cannot just walk into a shelter and euthanise them all, even if we wanted to. (And we don't).
You may not agree, and that's fine, but in my very experienced opinion I am absolutely comfortable prioritising dogs in these countries over UK dogs. I have also adopted amazing dogs from UK shelters, and I absolutely will again, but their fate is nothing compared to those in Eastern EU countries.