The medieval period still had ambassadors. Courtiers with the necessary skills could be sent to foreign courts to represent the interests of their monarch.

It wasn't necessarily as consistent as modern consulates, you might go for months or a year without someone from a foreign country attending the court. It depended on the circumstances at the time. Negotiation of a marriage between two Royal families could involve several commissioners from both sides working to hammer out the necessary terms (when Richard II of England was negotiating for a marriage to Anne of Bohemia for example, I don't believe it led to any permanent residence in her family's court for an English representative).

But once the marriage was completed those commissioners may return home, future negotiations could be carried out by different people.

We get closer to the idea of modern ambassadors around the 16th century. England played host to a permanent ambassador from Spain during Henry VIII's time, and long periods of having French ambassadors too. Likewise England had a permanent ambassador in France, especially during Elizabeth I's time. Anne Boleyn's father Thomas was famed for his diplomatic skill, and spent time representing the Tudor interests in Burgundy and France.

I'm less certain about how consistent it was before the 16th century. One of the good things about having so many foreign marriages was that the Queen could effectively act as an ambassador in her new country. They continued correspondence with their birth families, and their interests could be represented by envoys sent over by their families.