The browser extension sends the (public) user id of the user currently logged into X/Twitter to the marking/archiving server (xmark.cc) and this server uses the user id to obtain information about the top two posts of the user's X timeline. Typically top two means the two most recent, but if there is a pinned post, then it means the pinned post together with the most recent post. The information (post text, images, videos) is archived in the server's hard drive and a condensed version of the information is stored in the Bitmark (https://github.com/akrmn2021/bitmark) blockchain. The server sends the archived data to anyone who requests it through the cryptographic hash of the data that references it. Duplicate requests (referencing posts that are already archived) are not stored again in neither the blockchain nor the server.
All data that is collected by the browser extension is publicly available on x.com as well as other X archiving services.