I ran a Speedtest while watching a 1080 stream then did another after putting box into standby and the second test was about 5 Mbps faster. This would suggest to me that the boxes do indeed stop streaming in standby mode. Not scientific I know but seems conclusive enough for me.
Some say yes and some say no, haven't got around to checking the router lights when in standby, would probably be an indication one way or the other ( some say it stops streaming after 10 mins others say it continues to stream in standby and set to a FTA sat channel before standby) maybe someone can monitor router lights for 15 mins or so in standby
I normally record the streams rather than watch them as they stream and if I start downloading a stream and put the box in standby the download stops and I have to restart it when turning the box on again so I just leave the box on while downloading a stream
It stops streaming the second you put it on standby but be sure to leave it on a FTA channel before turning it on standby at night if you don't want to be unnecessarily streaming as it will start again in the morning when the box is updating the epg