This was a comment at  by me regarded to be off-topic by SimonPoole ("please stay on topic"). WTFPL licensed.
I hope these options for mitigating the problem and for understanding which issues play a role are helpful. SudduZ mentioned the "NSA thing" and "evilness of MS". This comment completes scai's answer and explains potential issues arising from that "evilness".
Also the user agent and referer (at least for iD and Potlatch2) is transmitted. Both are under the user's control and could be faked.
From the time/location relation of bing views and public edits Microsoft could infer which user name belongs to a IP address. You could weaken this relation if you would use caching of aerial imagery (which I think is done usually) and if you would delay your edits (using JOSM and locally saved .osm files – take care of potential edit conflicts). Also you could try to often get a new IP address from your ISP.
IP addresses could be hidden by piping the aerial imagery accesses through TOR or similar services (be aware that would create other problems).
Note that all our editors allow to switch off the bing aerial layer (JOSM even permanently).
Maybe it would be possible for osm.org to set up a simple relay for the bing aerial imagery, so that the IP addresses and other GET request details are not disclosed to MS. In addition with more pervasive HTTPS (currently it is only for the login and – optionally – tiles) this could improve the privacy.