The 350 million per week is correct in one sense: it's the gross amount we send to Brussels - actually it's a slightly understated figure as the average amount per week in 2014 was £360 million per week.
http://blogs.channel4.com/factcheck/factcheck-send-350m-week-brussels/22804
Now we get some of that money back from the EU, so the net figure is only about £83 million per week after all the rebates and returned money is taken into account.
I'd like to make two points about that though: firstly, £83 million per week is still a hefty sum; secondly, the UK government doesn't have much control over how the money is returned - and the amount returned could change over time.
I don't think it's wrong of the Leave campaigners to quote the larger figure: if I tell someone how much it costs per year to belong to my model flying club, should I tell them the figure they actually pay, or first deduct the benefits they get such as insurance charges and free coffee and sandwiches at the club meetings?
The Remain campaigners are just as guilty of misleading the public - they quote the smallest figure they can: some of the deductions they make are justifiable, others are more dubious.
The best way to state the facts would be to say that EU membership costs us somewhere between £83 million per week up to £244 million per week depending on which repayments and other adjustments you wish to take into account.