So these crawl torrent indexers and provide search functionality?

Do they also search private indexers?

They crawl the peer to peer DHT, private torrents have a flag so they do not get announced via DHT.

You can't search private indexers without having account on them.

they are not searching the indexers, they are searching the DHT directly.

Some private trackers do not set the "disallow DHT" flag, so those will be indexed as well. Most do, however, and it's impossible to scrape those without an account, yes