Being concerned by privacy issues when connecting to the internet, I know Twitter can monitor our activities if we don't logout and use some kind of firefox extension to prevent from being tracked. I would like to know if Twitter can monitor my browsing activities if I leave a Twitter client like Gwibber or Hotot open while I surf the web.
Nope. Twitter does not track anything while you are using a Twitter client. The client does not transmit any private browsing data to Twitter, it only sends and receives tweets, replies and PMs.