I moved my cat from a non-EU country to Germany. I did a bunch of research about this, but the most important thing in the end was to find a vet that deals with pet export (not all do, or are authorised to). The vet was able to point me to the official import/export agencies if I required further information. It is also possible that some pet export companies do everything, including having their own vets to take care of the medical side of things.
In my case, she required standard vaccinations, rabies vaccination, a special flea and tick treatment, microchipping, and pre-departure and post-arrival vet checks. What is required between EU countries, and for Romania and France in particular, can vary. Note that the timing of procedures can be extremely strict.
It may not be essential to have an "official" pet passport (I'm not even sure if there is an official one), but it is essential that all procedures and exams are documented. Especially if your dogs will be travelling in cargo, there are specific rules about which cages or cage types are acceptable that you will need to follow.
I strongly advise that you do not try to go this alone. Find reputable and authorised vets and pet export companies to ensure that everything is done as it should be.