In our "modern" system of equal tempered tunings, we encounter many compromises and trade-offs...but perhaps also some advantages.
In performing an equal temperament, the fifth intervals are made smaller by about 1 beat in 5 seconds, and the fourth intervals are made larger by a similar amount. Not enough to overtly disturb the purity of those intervals, but enough so that as you climb the scale, you don't encounter jarring mathematical asymptotes. In other words, in pythagorean (pure math) tuning, you get far enough up the scale to be "out of key" and suddenly you encounter 4ths and 5ths that are noticeably "out." The equal temperament spreads these mathematical "errors" across the whole scale, making each error almost vanishingly small.
One thing this does, however, is to make the third interval very much wider than it would have been.
In just (Ptolemaic 5-limit) intonation, we hear very clean, sonorous thirds. (to our modern ears, anyway!) In equal tempered tuning, we have been trained to be much more forgiving or perhaps better said, "tolerant" of varying degrees of sonority in our thirds. The standard equal temperament third (both major and minor) is a little bit sharp, thus feels a bit shrill compared to its 5-just sister...but this ambiguity has allowed us the ability...I might even use the word luxury...to take the third definitively and dramatically into interpretive territory, a place that the pure fourth and fifth can never be dragged, even kicking and screaming. I introduce into evidence "the blues." I can almost rest my case there.
In the context of the OP's question, the overall question of ensemble playing, the key is to arrive at consensus as an ensemble. The River of Thirds is wide and deep, and you all need to be in the same boat to cross to the other side!