Of course they do, in a way adding to the problem in some respects. I mentioned several factors. Migrant families particularly non-EU have much higher birthrates and can apply to have family brought over.
Non-EU?
I know they do but a lot do not and equally the British public suffer. I'll give you one such account. My local Hospital be brought in 2500 EU non-qualified nurses and covered all relocation and other costs. Each of them cost circa £25,000 in expenses in the first year. Then we have announced that the Nurses bursary is going to go? I think British people are quite rightly and justified in being peed off.
I can provide directly conflicting anecdotal evidence. I can understand being peed off with the scenario you describe but it seems clear to me that the issue isn't with the people it is that the bursary was cut (unrelated?) and when considered in conjunction with that, it's easy to point fingers and say "look at the money that was spent on immigrants and now the bursary is cut". That shouldn't be a reason to suddenly decide that immigration is the problem.
I know a huge number of EU immigrants working in highly professional fields - IT, engineering, manufacturing, transport, consultancy etc and they all contribute to NI and tax just like everyone else in the UK. I also know a large number of immigrants are employed in non-professional industries, retail, service industries, catering etc - they also pay their taxes and NI contributions.
In fact, in the wider picture and given the general work ethic of a lot of immigrants, I would go so far as to say that the proportion that do not contribute to society is lower than the proportion native people who are a drain on the system. In either case, they are both a significant minority in the grand scale of things.
You also forget that a lot of migrant workers send money back home that will not benefit the country. If you know migrant workers which I am sure you will say you do. You will know they send hundreds of pounds home each month.
One of my friends from Iraq sent back over to his brother in several lump sums to the tune of £35,000. He'd made several payments like this previously in the 10years he had been here. I know a fair few Polish blokes who rent here (2 families living in one house) who manage between them to send £1,000 a week back. Cheap area to rent, work industry type jobs (that have seen a salary contraction by the way despite business picking up). There would be tens of thousands of people doing exactly the same. You are talking tens of millions yearly at the very least gone from the UK.
They do pay tax yes. They may pay £200 a month in tax but are probably sending twice that out of the UK.
So? It's money they earned after providing a service to an industry and doing a job, paying their fair taxes and NI - are you saying they can't do what they want with it? What about British expats sending money back to Britain working tax free in the middle east? Should they not be allowed to do the same?
Will you be saying that the produce or products we buy should only be natively produced and that buying things from overseas is bad? Should we stop trading with other countries because after all, it's money gone from the UK...
edit: also, the figures in question are trivial when looked at in relation to the overall economy.