It's important to note that wash or no wash, mushrooms themselves already have a high water content so their ability to absorb more water is limited. Estimates I've found put the range between 75-90% water, depending on the type of mushroom. Cooks Illustrated, for example, puts it at 80%.
Sources such as Cooks Illustrated, Alton Brown, Harold McGee, Robert Wolke, and Kenji Lopez Alt have tested washing mushrooms by weighing them pre- and post-wash, and all found that mushrooms absorb a minimal amount of water that does not significantly affect cooking time. Both McGee and Wolke soaked the mushroom for five minutes - so much longer and with more direct contact than than a normal simple rinsing. Lopez Alt's testing, for example, found that the mushrooms absorbed only about 2% of their total weight which translated to an extra 15 to 30 seconds of cooking time. After testing, these sources all encourage washing mushrooms.
Several of the sources also observed that most of the additional weight is on the surface of the mushroom, which can either be patted dry with a towel or paper towel, or removed with a salad spinner, as other previous answers here have noted.
Cooks Illustrated suggests only washing whole mushrooms, as when cut they can become more absorbent. They also suggest that if you are serving the mushrooms raw, use a tooth brush to brush them clean rather than rinsing because rising can cause discoloration.
Washing should be only done just before cooking. Washing and then storing mushrooms can shorten their shelf life and they can become slimy on the surface.
Wolke notes that if your mushrooms are steaming rather than browning, it's more likely that your pan is too crowded rather than a function of having washed them.
Cooks Illustrated describes their process here, McGee in The Curious Cook, Wolke in What Einstein Told His Cook, Lopez Alt in The Food Lab (and also Serious Eats) and the best I could find for Alton Brown were transcripts from the Good Eats Fan Page. Both the the Cooks Illustrated and Serious Eats links also have more general tips on mushroom storage and prep.
Mushrooms should be stored in the refrigerator in a breathable container. A paper bag works great. Mushrooms stored in this manner should last at least 4-5 days.
For the record, the top of the fridge is one of the worst places to store foods. Depending on the model it can actually be a few degrees higher than ambient temperature, thus accelerating spoilage.
Best Answer
There's a great experiment on Cooking Issues that deals with this very problem. The general advice is to not crowd the pan because of the concomitant release of water; however, the guys found that doing this is actually beneficial because, although a lot of water is initially released, by the time the liquid has eventually evaporated the mushrooms have collapsed and cook far better in the oil that's left. Here's the full explanation:
What I do now is this:
Add the all the chopped mushrooms to a pan over a medium heat with some salt and water and cover.
The mushrooms steam and release and ton of their own water: allow to cook for a good 5-10 minutes.
Drain the mushrooms - reserving the released mushroom water - and dry the pan before returning it to a high heat.
When the pan is very hot, add a few tablespoons of oil (depending on the volume of cooked mushrooms) and throw in the mushrooms.
When they are nicely browned, deglaze the pan with the mushroom water
The reason for the two stage process is that I found if you wait for the water to evaporate naturally the residue from the dissolved solids in the water can stick to the pan and burn while the mushrooms are frying. By covering the pan initially, the water doesn't evaporate and can be siphoned off with any dissolved solids (that now can't get stuck to the bottom of the pan) and then reintroduced right at the end. As the pan is so hot it only takes a minute or two for all the water to evaporate and any new fond created by actually sauteeing the mushrooms gets used as well.