I'm not sure what you have heard, but as far as I know the only letters were sent from ports Titanic stopped at in Ireland and France, before it left for the North Atlantic. From out at sea, Titanic could only send wireless messages. Also, I believe the Cunard ocean liner, Mauretania, was the fastest "traveling" ship at that time, and for many years thereafter. Titanic had a post office, in the bottom of the ship. Of course, the mail would have been delivered in New York, and sent out from there.
The passengers, mostly first and second class, wrote letters in their cabins or in the libraries, writing rooms etc. They must have been able to get stamps somewhere, perhaps from the Purser. They put the letters into boxes placed handy to these places and the letters eventually went to the mail room.
Had Titanic reached NY the mail would have been taken off by a fast little steamer outside the harbour and would have been in NY before Titanic docked.
Joe is quite correct. Mauritania and Lusitania were much faster than Titanic. That's why loose talk about her trying to break records is hogwash.
About the speed - Mauretania's official record was 26 knots and it was said the Titanic could do 25 at most, maybe only 24. They were going to try the speed of 24 knots on Monday if the weather had been fine... but NOT in the night.