I always wash silk myself.
I wash most in the machine on a wool programme, in a mesh bag for delicate items, using Persil wool and silk detergent.
These items are silk tops, some scarves, DvF dresses (they're a silk jersey type of fabric) and so on. I usually just wash them with other items like cashmere, wool and similar delicate things.
Other silk items, like Hermes scarves, I wash in a small bowl (I worry that they will catch on edges in the sink), quite cool water with a tiny bit of the same detergent. I only wash one item at a time like this.
This is because I don't want to have them in water for too long as I worry about colour running, especially because some of my scarves are vintage and/or a mix of dark and light colours.
I roll the items in (white) towels until mostly dry, then let the items fully dry lying, rather than hanging.
I then steam the item (before I had a steamer, I'd iron on low heat with a tea towel between the item and the iron).
As Errol says, it's the colour running that you have to worry about.