When you use PuTTY, it stores a small amount of information on your computer, necessary for doing its own job. This information is stored in the user account of the user who runs PuTTY, so it is under your control: you can view it, change it, or delete it.
If you need to delete all of this data, you can use the -cleanup command-line option, as described in section 3.11.2.
PuTTY does not transmit your saved session data to any other site. However, you may need to be aware of the fact that it is stored on your computer. (For example, somebody else accessing your computer might be able to find a list of sites you have connected to, if you have saved details of them.)