DISCLAIMER: The following advice has been written down with utmost care and we have tested these methods ourselves; however we deny any liability for any results.
We strongly do not recommend putting plushies into a washing machine as the fur can get matted easily then. We've heard about quite some horror stories where plushies were ruined then, so don't do it. Although most tags say that the plushie can be machine washed, we strictly advise you not to do it.
Our preferred way is using a wet vacuum cleaner (must be able to soak up liquids, a regular one won't do) or carpet cleaner. In the US, there are some green colored models by a brand starting with B... sold that might come handy if you have several plushies.
Use a mild detergent. Some people from the US recommended Woolite, however we have no experiences with that. Avoid dishwashing soap.
If you have a wet vacuum cleaner, we recommend working with two buckets of water, one with and one without detergent. Plus a sponge and a soft cleaning cloth.
First, make the area where you want to clean slightly moist. Best if you go for an area the size of an A4/Letter paper or less. Then, use the vacuum cleaner to extract the water, but not fully. Apply a drop of soap/detergent onto the sponge and rub it into the area. Take as little soap as possible. Then, extract that with the vacuum cleaner. Now take the bucket with the water that will stay non-soapy. Wet the cleaning cloth and carefully rub it on the soapy area on the plushie. Apply the vacuum cleaner. Squeeze the cleaning cloth into the bucket with soapy water, then put some water from the other bucket into the cleaning cloth.
Repeat until the plushie is clean. We recommend to not do the whole plushie at once, but like half or a third so when the plushie lays down for drying, the moist areas stay up in the air.
This might sound complicated, but we found this to be the most gentle way of cleaning a plushie and it worked well for us in the past.
A carpet cleaner saves you a bit of work by applying and removing the cleaning liquid in one go. You may also be able to rent one of these at a local hardware store. Make sure to clean the nozzle throughly first.
Otherwise, we just recommend washing the plushie by hand in lukewarm water using a mild detergent.
Put some lukewarm water in a tub and add a dash of mild detergent. Put the plushie or the unstuffed skin into the water. Knead the plushie gently. The water will quite likely turn brown, this is pretty normal, even with barely used plushies. Change the water, add detergent and repeat until the water stays clear. Then, replace the water once more and do not add detergent drom now on. Knead the plushie again to make the detergent come out.
Make sure to wash the detergent completely out before you proceed. Hint: Push the fur and listen closely. If it doesn't crackle anymore like if you pushed into soap foam, the detergent should be out. Otherwise, change the water and repeat the step above. Dry the plushie outside on a warm summer day, but in the shadow. Do not put them into the dryer.
On a plushie up to the size of 75cm/30" length, you may leave the stuffing in for a hand wash. However, if you know how to do it, it would be better if you could remove the stuffing and put it back in when you're done. Wash the stuffing separately if needed.
While the plushie dries, ocassionally and gently brush the fur with a soft brush or hair brush.