Yes, and no.
To bid on a name you submit the
shaBid hash. This contains the name, bid value, sender address, and a salt. It is nigh on impossible to reverse. Your bid is not discernible from this value.
If you bid on a name using the
newBid method of the registrar contract on a previously started auction (
startAuction) it is basically completely anonymous as the only submitted parameter is this shaBid value.
If however you call
startAuctionsAndBid you have to provide an array of name hashes for the auctions that you are starting. An entity can use a 'rainbow table' to discern the names you are starting auctions for.
A rainbow table is a table of domains and their hashes. If a hash matches one in the database, then they know the domain. Some interfaces start auctions for 'fake' domains as well as the one you actually want to attempt to obfuscate the name you are truly interested in.
Regardless of which method you choose, you have to submit the amount of Ether (or more) that you are bidding. Some people send more Ether than the value of their bid to obfuscate the true value of their bid.
To win an auction, or to have your deposit returned you need to reveal your bid. To reveal a bid you again have to submit the hashed domain as well as the hashed bid value, and salt. At this point one can discern the details of your bid with relative ease.
Essentially.. to be as anonymous as possible, call
startAuction and then
newBid. If your domain is relatively obscure it may not be in the various rainbow tables going around.
You can find information about the ENS bidding process here:
ENS names are distributed by blind auction. The bids are revealed by bidders 48 hours before the end of the auction at which point new bids can't be placed.