Are we in silver border with this one? Because if I cast Gifts Ungiven and choose some cards from my teammates deck, they're going into my hand?

Not intentionally.

This only replaces who is searching, not the rest of the card effect. So your teammate's Evolving Wilds becomes "search your library for a basic land and put it on the battlefield under your teammate's control," and Demonic Tutor still puts cards from your deck into your teammate's hand. I don't think there's a good way to do this without giving control of the whole ability that includes searching to your teammate

What is the idea behind this card? What do you want to happen when your teammate casts a demonic tutor or cracks evolving wilds?

Is the idea that both of you can search? So that you get to tutor twice? Or do you want to be able to search your library and say "Hmm, all my swamps are exiled. Can I take one from your library Teammate?"

What was the original? I'm not sure, but I think you could add "if that card is put into their hand or library, put it into your hand or library instead". That would work with 99% of cards. Wait, and graveyard. And it would also need to specify that you shuffle afterwards. Ok, that's getting wordy.

Technically that's already implied by the rules. Anything that would put a card I own into your hand puts it into my hand instead, and ditto graveyard and library. (Rule 400.3.) So this lets my teammate use their Congregation at Dawn to find 3 of my creatures and put them on top of my library.

A lot of people don't know that though (and Unhinged didn't help), so it'd probably be better with words to say it, but they could be reminder text.

Oh, I missed that entirely. I thought it was a case of "if it can't go there it stays where it is". OK, then that would work just as intended, although I think it's a rule for edge cases, not one card texts are expected to rely on. But the reminder text could be a lot less precise if it just gives the general idea and doesn't have to spell out all the possibilities.

The intent is to allow your teammate to let you perform the search they would perform instead, in case it would be more beneficial to you than to them. Obviously it's not making a lot of sense. :(

Yeah it seems like replacing every part of that effect would be wordy, but i understand the ability. it's certainly general enough to be useful, but still feels kind of vague.. probs cus it hasnt been done... like at all...

maybe if you just play a bit of a new wording game, it will help, they sometimes do that with newish abilities.

What about "When your teammate would perform an action where they would search their library, they may have you perform that action instead"

right now it only covers part of the thing, but what you are saying makes sense....

Actually to be clear, if you just want to change who searches, than this is fine, if you want to change who gets the card after, well then that is where my wording is better/worse/something

and yeah this will break a lot of cards, unless you use a much more vague but technically legal wording.

Reminder text is also a nice idea, but the few specific cases where a card effect doesnt work with this and breaks something else would make that very wordy..

Mmm, I guess you could have "If an opponent would search their library due to a spell or ability, they may choose to have you control that spell or ability (and perform the search from your library, and have any further effects) instead"

But yowsers, horrible wordy.

"Anytime you are searching your library, you may stop searching, shuffle your library and copy the effect causing you to search your library. A teammate may gain control of this copied effect."

Doing stuff while searching is ok panglacial wurm but I don't think "effects" can be copied or "given" like this.