I'm blowing several moderations in this thread to make this point, but it's important enough that it needs to be made.
Almost no small sites were doing there own checkout process, those that are can use VAT MOSS in the UK (I'm sure equivalents exist in other EU countries) to avoid registering with ANY new tax authorities.
That is some combination of unrealistic and simply wrong.
For one thing, there are thousands of small/micro businesses -- actually, it could be millions, but no-one has accurate figures across all of Europe yet -- that are theoretically affected by the new EU VAT rules this year. The governments of the states individually and at European level completely dropped the ball on this one. They didn't even realise those businesses existed, as they have now openly admitted while panicking about the damage they've inadvertently done and now can't fix in time.
Many of those businesses have no idea they are now breaking the law. They haven't been told anything by any government authority, most of them don't have an accountant, and even if they do, even most accountants didn't see this one coming. Until this year those vendors probably weren't doing anything wrong because they were far too small to go above their state's VAT registration threshold while they sell their band's music or their knitting patterns or their e-book as a digital download. Under the new rules, there is no minimum threshold for registration any more: sell a 5 Euro download to one customer in any other EU state and you need to register either with that state's national tax authority or with your home nation's MOSS scheme.
Now, as soon as you're caught in the net, you are now into the practically impossible auditing requirements that require two non-conflicting data points to verify a customer's location. Even most payment services, marketplace sites and professional accountants can't cope with this yet, never mind your mother's side business selling PDFs of the local church's choral arrangements for a token charge to raise some money to fix the church's broken window with a simple web site and a PayPal button.
Of course, all of this is being done because you now have to charge VAT at your customer's state's rate instead of your home state's. You almost certainly don't know that yet when you first show a customer the price, because you don't have the required multiple non-conflicting data points to satisfy the audit rules. In practice, that means you probably can't comply simultaneously with both the new EU VAT rules and existing consumer protection legislation that requires prices for B2C transactions to be advertised tax-inclusive, and you will be breaking some law one way or another even if you make a good faith effort to comply.
Even if you are doing almost everything yourself and can comply with the audit and advertising rules, you then need to register with your national data protection authority because you'll have a requirement to maintain personal data about your current and past customers for 10 years and be subject to audit at any time by any of 28 different tax authorities. Or you don't need to register, because the data falls within an exception. Not even national governments are giving consistent information on this point so far.
Once you've registered with everything you need to register for, you might then wind up putting up your prices for everyone, because you're now effectively prevented from relying on the VAT threshold (though to be fair at least some EU states have taken emergency steps within the last few days to address this problem).
On top of all of that, you now have ongoing reporting obligations that probably require you to understand a minimum of two VAT-related returns (one national in your home state, one MOSS) so you can look forward to either spending several days becoming an expert on your national tax law or paying a substantial amount of money to professional accountants and hoping that you can find one that understands the new rules themselves.
I could go on (and I have, but in more useful forums and in official letters to my representatives) but if you think this change is some trivial detail that doesn't really affect anyone, you are either trolling or severely ignorant of the real implications. Please go and read euvataction.org and learn something, or just look at how many businesses are giving up according to #vatmess on Twitter. Numerous small businesses have stopped supplying to other EU nations as a direct result of this (or shut down completely) and I've seen more than one professional accountant openly saying that if you make less than four figures a year on your side business you might as well shut it down because the compliance costs are going to eat literally all of the money you ever made.