My US company offered me the prospect of doing my PhD (in public health, public policy, international development or something similar) at a European university while retaining my employment. They would also pay for the PhD. My thesis would be based on a research project I was already going to pursue in my work. I already have a Masters in Economics. Would a university be okay if I only worked out of the university for the first and the 2nd and 3rd year was finishing my research in the USA? Thanks!

I did this and it wasn't a problem for my supervisor. We met up once a year in person and the rest of the time we kept in contact using Skype. However, it proved impossible (for me) to do a full-time PhD and a job so I had to switch to a part-time PhD to finish in time. I also missed out a lot by not being on campus. (It was lonely!) But realistically it was the only way for me to do it.

I think timefortea's point about it probably needing to be p/t is very true. It is also going to be more like the equivalent of 4 years however you do it than 3 realistically. However supportive your work is, stuff happens that pulls you in different directions and it's always the PhD that is lowest priority for them. Distance learning is viable but you do miss out. I think another thing that you might want to consider is career outlook. This way would be great to help you perhaps advance in your current career, but might be sub-optimal if you wanted to move into academia.