Do you have duties other than monitoring and reacting to Skype chats? If yes, it is virtually impossible to manage all duties considering the high frequency of chats. You may need to talk to your boss about your job function.
With regards to excessive trash chats, they are distracting and hampering productivity. You may want to escalate to your boss. First, create an organization culture of minimizing trash chats. Second, encourage staff to create chat groups to talk about matter unrelated to work.