What are the key regulations governing private company stock ownership in Europe, and how does this compare to the regulatory landscape in the US?

We’ve got the equivalent of America’s SEC and FINRA in each country. You only need to check, depending on the level of regulatory approval you have, whether these agreements are passportable or not in all the EU member countries. 

What we are aiming for is something that is passportable across those 27 countries. In the UK, you need a separate setup to actually be able to operate, but for now, the country hasn't diverged from the standards present in continental Europe.