If an effect says "You may search your library . . . If you do, shuffle your library" or "You may search your library . . . then shuffle your library," you can't choose to search, so you won't shuffle.

If an effect says "Search your library . . . then shuffle your library," you shuffle you libraries even though you can't search.

Effects that instruct you to reveal or look at cards from the top of you library will still work. Only effects that use the word "search" are affected.

Since players who haven't paid {2} can't search, they won't be able to find any cards in a library. The effect applies to all players and all libraries. If a spell or ability's effect has other parts that don't depend on searching for or finding cards, they will still work normally.

Paying {2} to ignore Leonin Arbiter's effect is a special action. Any player may take this special action any time they have priority. It doesn't use the stack and can't be responded to.

Paying {2} doesn't let a player pick up a library and search it — it just allows them to ignore Leonin Arbiter's effect that turn. That player can search a library only if another spell or ability instructs them to do so.

If a player pays {2}, that enables only them to ignore Leonin Arbiter's effect that turn. Each other player is still affected by it.

A player who has paid {2} may search any library when instructed to do so that turn, not just their own library.

Once a player has paid {2}, they may search libraries that turn as many times as they are instructed to do so.

If there are multiple Leonin Arbiters on the battlefield, a player must pay {2} for each one before being able to search libraries that turn.