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.
Soak them for a while in warm water, and you'll be able to use them, yes.
However It is equally important to know that you should reserve the liquid for its essential mushroomness, and yet also that said reserved liquid should be run through a coffee filter to remove grit.
Best Answer
My advice: ditch the soaking liquid.
Here's what I just tried. I divided my dried mushrooms up into 10 bowls: 5 with dried chanterelles and 5 with dried porcini. I added equal amounts of water to each at the following temperatures 10°C (directly from the tap), 40°C, 60°C, 80°C and 100°C (or as close as I could get).
After soaking for 15 minutes I sampled the soaking liquid from each bowl.
All samples tasted astringent and bitter. Naturally the taste was less pronounced for the colder liquids, but not so much as I had expected. The liquid that the porcini soaked in tasted somewhat better than that of the chanterelles. None of the soaking liquid tasted good enough that I would add it to a stock; I was almost gagging when the experiment was complete.
I wrung as much liquid as I could out of the rehydrated mushrooms and used them to make a vegetable stock. With the flavor of the soaking liquid fresh in my mind I could still detect a note of bitterness. This was easily dealt with using a little salt.
The bitterness from the soaking liquid, I know from past experience, cannot be completely removed with salt.
Update: Looking at messages from an online forum here that deals with the same problem, I thought I'd look at some suggestions made there.
There was an amount of dust in the dried mushrooms I purchased. However, I rinsed the mushrooms before starting the experiment described above, so dust doesn't account for the bitterness.
I'm not 100% sure about this, but wouldn't bruising change the color of the mushroom? I didn't notice any discoloration. Anyhow this is a quality issue and I've more to say about quality later.
The water in Stockholm is neither chlorinated nor laden with minerals (I have had the same kettle for years and there's not a hint of mineral deposit inside). This is not the problem.
Quality seemed like it could be an issue to me. My impression is that supermarket dried mushrooms in Stockholm don't have a high turnover and could have been on the shelves a good while. The mushrooms I used in the tests above had quite a lot of dust in their packages and that surely isn't a good sign.
Today I looked around for something of visibly better quality and found cloud ear and shiitake mushrooms in the local Asian store. These mushrooms look pretty good and their packages contained almost no dust. The dried shiitake were vacuum-sealed in a package together with an oxygen scavenger.
I soaked these mushrooms at 10°C and 80°C for 15 minutes. Cloud ear mushrooms don't have a strong smell and the soaking liquid didn't taste of much after soaking. The little taste I could detect was slightly astringent and not particularly pleasant. Incidentally, cloud ears don't rehydrate well in cold water and were still quite solid after soaking.
Dried shiitake, in contrast, have a strong musty smell and the soaking liquid was relatively strong tasting. I did taste some pleasant mushroomy flavors and for a time I had to weigh whether I liked the taste or not. Unfortunately these flavors were mixed with some definitely off tastes, among them the familiar bitterness. In the end I couldn't stomach more than three tablespoons of the liquid.
I had a notion that mushrooms soaked in cooler water might retain their mushroom flavor whilst leaching their bitter taste. To this end I drained the mushrooms that were soaked cold and added boiling water to them. After a further 15 minutes I taste tested again. Comparing this second soaking liquid to that of the shittake initially soaked at 80°C, the flavor was much more dilute. I'm sad to say that mushroom flavor is lost even when soaking cold.
My advice for now remains the same: discard the soaking liquid. If that strikes you as wasteful I would at least urge you to taste a tablespoonful of the stuff before you decide to add it to your broth.