Most live chat systems actually show the agent in-progres messages. I've integrated a few of the available live chat services myself and they all do it by default. It quite often helps the agent to understand what the customer is trying to get at/what they want, you'd be surprised quite how many customers will sit writing an essay explaining a common issue that the agent can answer in seconds.
If you don't want the agent to see what your typing, do it in another window and copy the text over after. They will most likey realise you are doing this but at least you have time to compose your answer before they see it. This applies to all sites with live chat, not just bitstarz.
That is correct. Nowadays this is a very common feature and i think it correlates with usual privacy t&c. I work for the customer support of one of germany's biggest broadband provider and this tool was implemented 1 or 2 years ago. It's very useful and we never had any trust issues.