You are going to get answers that say "yes". Those answers are correct. You are going to get answers that say "no". Those answer are also correct. In other words, it's complicated.
RM will send data about the person in RM to the site to ask the site if there are any hints for the person. The site will not capture or store the data. But the site will use the data temporarily to look for hints. If any hints are found, the hints will be reported back to RM.
The way hints are identified is very different for each of the three sites, but that's not what you are asking about. What you are asking about is your RM data. A small bit of your RM data has to be sent from RM to the sites and kept by the sites at least temporarily while the sites are looking for hints. The way FamilySearch hints and ancestry.com hints work, it appears that that data you send is kept only briefly as the hints are being processed, perhaps only a few seconds or even less. The data sent in the query then appears to be discarded after that
The way myheritage.com hints work, it appears that the data in the query is actually kept around much longer. It's just the data for the one person for whom the hints are requested that is kept. And the data appears to be kept in a form where it's not in any database that any other user can get at. But it seems to be kept around. The reason is this must be the case is that when you get a hint from myheritage.com, it appears on a Web screen in a browser and the screen includes your data that was sent and also the data for the hint. The Web screen is constructed by myheritage.com. Therefore, in order to place your data on the screen it has to have your data, if only briefly.
But if you save the URL for that Web screen, you can wait weeks or months and the same URL will give you the same screen, including your data. So myheritage must be saving that screen somewhere, or the data to reconstruct that screen somewhere. The screen is not stored in a myheritage.com database and it's not available to any other user, but myheritage.com has to be saving the data some how or other or else it couldn't send you the same screen again months later. I wouldn't say that myheritage.com is capturing or storing your data, certainly not in its databases. But it's saving enough of your query to send your query back to you later.