Vaguely. Basically each CA will run one of these. Relying parties (browsers) will need to fetch the Merkle tree heads periodically for at least the CAs that sign certificates for the sites the users browse, or maybe all of the WebPKI CAs rather than just those that sign certs for the visited sites. There's on the order of 100 CAs for the whole web though, so knowledge that your browser fetched the MT heads for, say, 5 CAs, or even 1 CA, wouldn't leak much of anything about what specific sites the user is visiting. Though if the user is visiting, say, China sites only, then you might see them only fetch the MT heads for CN, and then you might say "aha! it's a Chinese user", or something, but... that's not a lot of information leaked, nor terribly useful.
I think most folks involved are assuming the landmarks will be distributed by the browser/OS vendor, at least for end-user devices where privacy matters the most - Similar to how CRLSets/CRLite/etc are pushed today.
There's "full certificates" defined in the draft which include signatures for clients who don't have landmarks pre-distributed, too.
The privacy aspect isn’t that you fetched the heads. It’s that you are sending data that is likely usable (in combination with other data being leaked) that websites can use to fingerprint you and track you across websites (since these heads are sent by the client in TLS hello)
The heads don't change often. I think we should list all the metadata that gets leaked to see just how many bits of fingerprint we might get. Naively I think it's rather few bits.
Sure, but every few bits is enough to disambiguate just a little bit more. Then each little trickle is combined to create a waterfall to completely deanonymize you.
For example, your IP + screen resolution + the TLS handshake head might be enough of a fingerprint to disambiguate your specific device among the general population.
