But if you do it yourself, do not use your personal information like the form will ask you to. All the information of the DMCA form is sent to whoever is hosting the content, generally a bad idea unless you have a blackmail fetish you want to live out.
would never suggest someone do this of course

but one
could conceivably use fake info, pirate sites aren't gonna check that kinda thing whether they choose to counter or not
It is very common. I just let it be, personally, because as fast as I can send a DMCA notice there will be 4 more streams uploaded. It's like playing whack-a-mole. But depending on what site you work on, you can contact them about helping you proceed with sending DMCA notices to each site your broadcasts are uploaded to. If the site doesn't offer such a service, you can do it yourself, but the way to go about it is a bit out of my ballpark. Google can probably help.
yeah, it's def better to be strategic. everyone has their own system but something to consider
@bosskaan is if the recordings are actually short enough to provide free promo. in that case it might be a choice to just leave them be, as infuriating as the theft is
re your suggestion to seeking site assistance, that's smart advice for sure. not just it's great if the OP's site offers it, but cuz some sites actually provide both free chat and private show recordings (not live streams, past recordings) to their affiliates to use in promo. there's no use DMCAing an affiliate site with your content if they've been provided with that content by the site itself (which as per contract terms actually has license to use the streams on their platform however they wish, including incorporating them into the affiliate program)
re sending DMCA's oneself, to actual pirate sites, some sites actually have a DMCA removal link (usually in the site footer) or a contact email address. use that to send a basic template (easy to find with a quick goog search) where you copy in your info and the relevant URLs, both infringing and original
some sites comply, but many sites will ignore. if you want to amp it up you can also DMCA their host and whatever advertising platforms they use. marginal success generally with this, but if it's really p!ssing you off worth a try
either way, you can DMCA google to get the URLs to your pirated stuff removed from google search. this makes your content a smidge harder to find, and can also potentially ding the site's overall rankings in search. for that reason always ensure it's actually pirated content, and not material actually provided to the affiliate by the site you work
pirates suck, good luck OP