It is estimated that nearly 30,000 German mercenaries fought for the British over the course of the American War of Independence. However, these men were not independent soldiers of fortune, but enlisted men in the service of various small German states, whose services were rented out as a unit by their rulers. This practice of purchasing the service of another state's army was relatively common in 18th century Europe. Of these 30,00 soldiers, some 19,000 of these men were from the landgraviates of Hesse-Kassel or Hesse-Darmstadt which is why they are colloquially known as "Hessians". During the terms of their service, they would have been paid their regular wages (at least in theory...) with their prince pocketing the subsidy paid by the British crown to their home state.

I don't know off the top of my head how much Hessians were paid, but considering how relatively small and poor Hesse-Kassel was, a safe answer is "not much".

It is known that between death and desertion, only about half of the Hessians returned to Europe after the end of the war in 1783. Exactly how many stayed in the new United States after the war is not precisely known, but generally estimated at around 5000-6000 persons.

For further reading I recommend Brady Crytzer's book Hessians: Mercenaries, Rebels, and the War for British North America, which is a fairly quick and easy introduction to the topic. If you want to get into the nitty-gritty of how Hessian soldiers were recruited and organized, check out Charles Ingrao' The Hessian Mercenary State. It discusses how the German state of Hesse-Kassel recruited, trained, and paid it's soldiers. If you want to learn more about how Hessian soldiers sometimes struggled to assimilate into American society in the early Federal period, consider The Trial of Frederick Eberle: Language, Patriotism and Citizenship in Philadelphia’s German Community, 1790-1830 by Friederike Baer.

Follow-up: I was under the impression that it was primarily Britain that used the services of German mercenaries during the American Revolution. Other than privateers and of course the various groups of Colonial citizens that made up the army, what variety of mercenaries did the colonies employ? And were the Hessians mercenaries in the way we’d think of them nowadays?

In America Goes to War Charles Neimeyer tells us that roughly a quarter of the 30,000 German mercenaries shipped to North America settled there.

Answering the question of how many would have fought for the Thirteen Colonies, Neimeyer goes on to look at Brunswickers, where the German archives have records on 1,700 of the 3,015 who did not return to Germany. Of these, at least 83 served with American combat forces. This is just under 5% - so extrapolating (dangerous) gives us a rough estimate of 375 Germans defecting and fighting for the Thirteen Colonies. Far more would have simply deserted and settled in America.

Let's go on to why they defected from their paying job. Under convention of the time, prisoners of war had their upkeep paid for by their own side - e.g. a British soldier captured by the Americans would have his food and clothing provided by the British. The British refused to do this for German mercenaries, and so the mercenaries often chose to desert or enlist with their former enemies.

Source: Charles Patrick Neimeyer: America Goes to War: A Social History of the Continental Army