Do companies know that I monitor their website?
As soon as you open a web page on a server, some data are submitted to the server (for example your IP address, the User Agent, etc). This happens independently of the software you're using to open a page (Internet Explorer, Firefox, Chrome, WebSite-Watcher, RSS feed reader, etc).
By default, WebSite-Watcher is identified as Internet Explorer, so a webmaster (or server) will not know if you open a page with your Internet Browser or with WebSite-Watcher. This identification string (User Agent) can be changed in the program configuration or per bookmark in the bookmark properties.