The operation of a PuTTY network tool is controlled by its configuration. This configuration is obtained from:
The defaults built in to PuTTY do not tell it to save log files, or specify the name of any network site to connect to.
However, if PuTTY has been installed for you by somebody else, such as an organisation, then that organisation may have provided their own default configuration. In that situation you may wish to check that the defaults they have set are compatible with your privacy needs. For example, an organisation providing your PuTTY configuration might configure PuTTY to save log files of your sessions, even though PuTTY's own default is not to do so.