I do so many searches for my 'availables' list, I can really compare these things; I've found:
Godaddy, it seems, makes visible your searches. If someone searches a word, on the results page you'll see, down below, a 'more options' area, that seems to be a mixture of owned names for sale, then your keyword that has been auto-joined with the same kinds of secondary words each time like 'best', 'free', 'new', etc. Many registrars do this.
Then it also shows some names, with your keyword, that are obviously other people's searches... they're just too random, way out, to have been autoscripted by GD. I don't know if GD themselves would nab any of your searches, I doubt it, but that's irrelevant since anyone can see your search terms.
In my experience, my searches at Godaddy result in the most of my searched names taken, and taken the fastest, of any other place I've ever searched.
I have a few dozen names there, but don't use them much anymore due to that and other reasons.
Namecheap searches, for me, result in a lot of my searched names taken also, and quickly after my searches. Again, I wouldn't think Namecheap gets them, that just doesn't make sense or ethics and I'm sure they don't do that. But someone's seeing my searched names when I go through them.
I do, however, love them as a registrar, and keep almost all my names with them, except the best ones, which I keep at Moniker.
With Moniker searches, I've had a much smaller percentage of searched names taken, and results are nebulous because I don't search there often. They, like GD, seem to have a more options area where others' searches are made public when you search a specific term, but again, I don't do searches there enough to say anything for sure.
I use dnaz for secure searches. It has the occasional glitch and will show a name as avail when it isn't... but I've never had a dnaz-searched name be taken shortly after my search. Just never happened. Dnaz is short for domain name analyzer, a small program you download to your computer.
Note that I haven't tried iwhois.com, but may give them a shot soon
To summarize: I'm not pretending to know what the registrars do with our searches, but when I see that I can search thousands of names through one place and many of those names get regged within a couple days, then search thousands of names through another place and none, or only a couple, names are taken by chance... that makes things very obvious, the results speak for themselves: It's not about 'coincidence' or 'chance' that others just regged those names, except in a very small percentage... in one place, they're seen and taken by whoever can see your searches, in another place they're not seen.
I have no problem with the registrars about this. Just need to be careful where you do your searching, if it's for names you might not reg right away.