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 typically release water if they are overcooked and also if salt is added too early in the cooking process.
When cooking mushrooms, cook them on a relatively high heat until they have just developed some colour. at the end of the cooking process add your seasoning.
Best Answer
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.