There are two schools of thought here from what I’ve seen in pro kitchens, in person and on TV:
1. Don’t wash your sifter. Instead, give it a good shake and put it away. If you can keep it from getting wet or coming into contact with anything but flour this makes pretty good sense and is probably the most popular choice.
2. Get a top-grade stainless steel sifter, shake and hand wash (cold water keeps any flour from getting pasty and tough to remove), spraying with a high-pressure hose if you have one. I’ve often seen pastry chefs put their sifter in the oven on 300 degress or so to cook lumps of flour they can’t easily remove before washing.
And then there are those that’ll tell you to just get a sieve (often confused with a sifter), which is a lot easier to clean and potentially space saving, since it’s multi-use.
@CharlesT: thanks for the stainless steel chemistry lesson - I had no idea the brown stuff that prompts me to toss out expensive stainless steel equipment is not actually rust and completely removable.