Even up through WWI (albeit more so on the Eastern Front), much of scouting was done on horseback, then reported back to command. The telegraph, telephone, radio, and airplane began to change things tangibly only during WWI, so I'm not sure where you're pulling '1850' from as the date when reconnaissance changed forever.
As for size? It varies. Light cavalry, parties of five to five-hundred, depending on the main unit size and the expected size of the enemy. Much of the combat engagements fought throughout history were skirmishes between opposing scouting parties. I'm paraphrasing LSCatillina here but for the most part, medieval warfare comprised of sieges and the actions therein. It was the duty of scouting parties to identify the positions and routes of the enemy armies, to prepare their supposed targets for a siege when they should happen to arrive.
Armies very rarely spent themselves fighting each other---soldiers and the treasure needed to pay them were precious, so pitched engagements really were only necessary when a campaign of successive sieges/sacks of enemy towns wasn't an option, or could only be prosecuted once the enemy army was routed.
Scouts, thus, provided the extremely necessary information of "figuring out where the hell everyone is, and where they're going". This could be done a number of ways:
-Scouts can ride ahead of a main army to locate the enemy, so that the army can meet them in battle/avoid them;
-Established outposts along a frontier can send riders -back- to a main army to achieve the former goals;
-A continuous border patrol can achieve the former, as well (etc.)
As for the actual reporting itself, that depended on the distances involved. Equipment and tactics depended on the army doing the scouting.