I'm not particularly worried about other applications on my computer listening to the clipboard. But I 99% of the time I'm pasting into a webpage in Chrome or Firefox. Can any open tab sniff the clipboard passively?
No, webpages get to the contents of clipboard only after explicit user interaction.
On the other hand it does not work this way in the other direction. Random web pages can manipulate your primary selection and overwrite it with random garbage (this primarily happens with various attempts to make copying stuff from the page more "convenient", pretyy comonly resulting in state when it is simply impossible to copy said thing into say rxvt directly). It is somewhat ironic that chrome's address bar uses some magic to prevent this from happening, while the same magic is not applied to websites.
