At worst you're giving the attacker a hash target to try brunting. He still has to brute it, and that takes time. Select your plaintext from a large enough keyspace and it's astronomical time.
I'll need to review their policy more closely, but DDG claim fairly minimal tracking. At best someone might be able to correlate hash lookup with some IP space. That's a long way from handing over passwords. And as I already indicated, you could cradled the queries to make the search space much larger.
When you search for 'sha1 foo', that query ("sha1 foo") goes up to the server. They know your password is "foo" and that you're attempting to "sha1" it. They don't have a hash, they take that data and perform the hash, then send that down to you.
I guess I'm just too damned used to using systems that, you know, have useful tools installed locally (or can get them there really damned fast). Including SHA1 and MD5 hash generators.
And I was all worked up to tell you how wrong you were still being.
All because I couldn't fathom the possibility let alone reason anyone would need a third-party site to compute their hashes for them.
Silly me, my error.