Were letters from Europe or European colonies shipped on merchant vessels along with cargo? Or was there a distinct postal service? And was it possible for people from the colonies to send a letter to a country other than their respective mother country (say, from post-1768 Spanish Louisiana to France and the other way around)?
Winifred Gallagher touched briefly on this question in her book How the Post Office Created America. The answer I will provide is concerning mail immediately after the establishment of the US Post Office in 1792.
Before universal postal standards in the Western world, America would correspond with Europe by a very shaky system. First, an American citizen would write and address a piece of correspondence simply with the recipient's name and the town where they resided. This piece of correspondence was then sent either to the customs house or through the burgeoning US Post Office to then be placed onboard a ship heading towards the closest European port. They would then trust that the mailbag would then reach each country's postal service (or similar institution) where the deliverer of the mail would then have to determine exactly to whom the letter should be delivered.
It mostly worked, especially between American and British correspondents since the British post office was an established and standardized institution. The timeframe between writing the letter and receipt of the letter back would vary according to accurate delivery and the schedule of ships heading back to the closest port in the New World.
Keep in mind, though that mail was routinely lost. Either lost in the States, lost at sea, lost upon unloading, lost in the receiving country's mail system, etc etc. There was also the issue of correct postage. How could anyone in the New World know what the needed postage would be when sending a letter to Belgium, Germany, or Spain? Therefore, the sender would affix as much as they thought necessary to the letter.
The other method, used mostly in business correspondence, would be to send the letter in the possession of a trusted courier or acquaintance who was taking passage on a ship heading back to Europe. This correspondence was then disseminated to the recipients via hand delivery.
In all, it was quite hectic and not altogether trustworthy until the establishment of universal postal standards. It was not uncommon for multiple letters to be written to Europe over several months restating the previously letters' contents with new information tacked on.