I'm not following...there aren't 2 parties. There's 1 party, the people sending a take down notice. The question is why aren't they sending the same take down notices to all search engines which have the material in question?
Quite possibly for the same reason that many, if not most (? I've no idea, really, but certainly plenty of), SEO companies consider Google to be the only actual target, because Bing just isn't popular enough to spend the time on.
If there was a search engine out there that had 10 uniques per month would you expect people to care enough to send take-down notices to them? So where do you draw the line... for some people, you draw it between Google and Bing, not below them both.
Maybe they have and Bing just prioritizes requests differently?
This is interesting though, when I saw the headline about Microsoft making so many takedown requests to Google I assumed the items to be taken down were found by the Bing team, but it seems that isn't the case.
Maybe it is and the Bing team only uses Google to find the links. (lol) Actually didn't Microsoft force their employees to use only Microsoft products? Sometimes it's not fun to eat your own dogfood.
As far as I remember, it was them asking not to buy iPhone. Nothing more. They can't force their employees to use only Microsoft products. They'd be down in a day!
The 2nd party is the team at the search engine that has to take the link down. Maybe Google processes the notices more efficiently. Or maybe Bing prioritizes non-MS takedowns, because they are at greater risk of suit from them.
If there was a search engine out there that had 10 uniques per month would you expect people to care enough to send take-down notices to them? So where do you draw the line... for some people, you draw it between Google and Bing, not below them both.