Read the twitter documentation for developers, starting here :
https://dev.twitter.com/rest/public/search
and then ask around to find someone who does "web programming" who can automate the twitter-search for you.
At least, skim this documentation.
To do anything useful, you need to register with Twitter's development program, however simple your task might be.
They'll give you (for free) two pairs of looong anonymous username/passwords, "tokens". One pair of tokens is for your registered "app"- the things you, or your little program, wants to search for. The other pair is for your account. This way you can configure/use many search-apps, and enable-disable them individually.
You'll see quickly: Asking for followers, favorites etc of users is possible with tokens, (username/pw) only. Moreover it is rate-capped. This means you can ask for the details of 15 followers max, then you must wait for another 15 minutes. So Twitter forces you to think ahead and think thoroughly about what you want to get from their search interface.
You can do mass-downloads but these are more complicated, and less specific (you must do the filtering).
For facebook it's a similar procedure, I guess.