Hmm, I think it depends on the two countries. We were flying from Italy to the US, and we had to get a good health certificate and rabies vaccination certification through the official Italian authorities. Both certifications only were good for six days, so we had to time everything at the last minute in case the temperatures on the east coast went up (thankfully they didn't) and we had to delay our flight back. Interestingly, the complicated certification paperwork had to be done to get our dog out of Italy (even though she is an "American" dog), but all they needed in the US were health and rabies certificates.
Hope this helps somewhat. I got all of the up to date information on the Department of State website. The Italian websites had conflicting information.