Yes, some countries require quarantine. What you do is first select which countries you wish to ship to. Then you find an export facility that is willing to freeze for those countries (since each is different, some are more complicated.... some facilities aren't able to qualify to ship to all countries). Then the facility will tell you, the stallion owner, what types of tests you will need to undergo. Some countries require an initial set of testing, then quarantine for a length of time, then retesting, before the semen can be qualified. In the mean time, collections can still be used for domestic shipping, as well as some less strict overseas countries. As you can imagine, it gets quite complicated keeping it all straight on the paperwork. Once everything is clear and done, the USDA/State Vet has to approve the shipment.
Generally, 60 days residing at the freezing facility is all you need to store up a lifetime supply of export semen. Unless your horse is REALLY popular
In which case you may be profitable to freeze 2 months out of every year!
The facilities that offer these export services are more than likely some of the top facilities in the country... so leaving your horses there in quarantine is not that big of a deal. And visitors are allowed, usually, so you can come see your horse. Its not like an import/export facility for horses, semen is much different. Besides, you don't do this during breeding/show season... usually you freeze for export in the fall/winter.
The most efficient way is to load a large wet/dry shipper full of multiple stallions, and ship it overseas to a broker. Its MUCH cheaper that way, to share shipping. Plus, you generally find it more cost effective to BUY a brand new shipper ($1500+) to guaranty it's sterile, rather than cleaning one out and expecting it back.