To share some recent experiences (as someone who has to notify).

Eurostar - quick look at my passport both ends only. No questions, no 'discrete' checks. Same in both Paris and Brussels.

Eurotunnel with car - quick look on way out, nothing more. Lot longer process on way back - passport obviously flagged on UKBA system and the guy didn't really know what to do, so a long time asking all the usual questions - where had we been, how long away etc. and he wrote all the answers down by hand on paper. Not a big issue as my fellow travellers aware of the situation

Do you know if you can live abroad on notification (sex offenders)???

In Europe you can ( for the next 2 years at least ). if you are seriously thinking about living abroad I'd go now 

Being able to move there is the easy part (for now). But the question is what kind of employment can you take up? Do they do background checks for most jobs in Europe?

I used to live abroad, while I was on the register.

The local police I deal with in the UK had no problem with me moving. The only logistical thing to do was to do a foreign travel notification whenever I returned to the UK. If I didn't know the next time I would be visiting, they simply leave the return date blank with a comment.

I moved to a capital city and got a standard office job where English was the main language, but it obviously helps a lot to learn the local language.

With regard to checks, most EU countries do not have the austere level of checks that the UK has. In the country I was in (a western European country), employers are forbidden point blank from asking about criminal records unless you are applying for a job working with children, vulnerable people, security, etc. There was no "basic disclosure" or anything like that. Most EU countries are much more respectful of privacy.