I've been a drummer for 30 years, and have played with countless other musicians, most of them string players. I will notice when playing with multiple guitarists, or a guitarist and a bassist, that they will tune to the same tuning. Are there ever times that guitarists/bassists would want to be in a different tuning? I'm asking because I've never experienced it.
There are two different meanings of »tuning«.
Not related to any instrument: Here we define a pitch for a certain note (most common the A note). A common tuning standard is A tuned to 440Hz. Another is A tuned to 443Hz. When two guitarists play together they have to make sure that the A at one guitar has the same pitch as the A at the other guitar. There exists no case where one player tunes his guitar to 440Hz and the other player tunes his guitar to 443Hz. This would sound terrible.
2) Tuning system
Related to guitars/string instruments: The order of notes the different strings are tuned to is also called »tuning«. Guitars are most common tuned to E standard (E A D G B e). But it is totally fine if one guitar is tuned to open A (D A d f# a d') and the other to E standard. They don't have to be in the same tuning in order to play together. But an A on the one guitar must have the same pitch as the A on the other guitar.
Same reason, really, as to why some guitarists need/have to use capos. If I'm playing a song in concert B, I'm happy doing that. Another guitarist might prefer to capo on 2nd fret and play it in his mind, in A. Generally, all guitarists will tune to concert, but if one decides he'd rather play a tone lower, for better sound, etc., then he'll tune differently and compensate. It may well be that you've already been party to this, but not even realised - there's no good reason why you should - everyone is effectively in the same key, so it isn't apparent.
If two guitars are played in the same tuning, and the musicians use the same chord shapes and scales, this can be uninteresting. For example, if both players are using open chord shapes, they are probably playing the exact same chord notes in the same voicing.
One way to make it more interesting is for one player to, for example, play up the neck while the other plays at the nut. This changes the chord voicing, adding interest.
Another way to achieve similar results is for one player to play in a different tuning. This likely changes the voicing of even open position chords, so that even if both player play near the nut using open chords, they won't sound exactly the same.
If by "Tuning" you mean "pitch" as "A - 440Hz" then no. All the instruments, not just guitars, but piano, fiddle, bass, synths etc. need to use the same "baseline". The standard is 440Hz. You tune to the instrument which can't change its tuning easily (ie. grand piano)
If you mean the order in which the strings are tuned, then standard is EADGBE. There is no need to have everyone tuned the same as long they adjust, in fact, it will probably sound more "interesting" if some players tune differently. For example, slide player will tune to open chord while rhythm player will stay in standard (or play in higher position with capo).