It appears to me that if you're in stealth mode, and attack a creature, and you're more than 3 cells away, the creature makes a wisdom based perception roll to detect you. This appears as the DC in the log. You make a dex(stealth) check to beat the DC. If you have pass without trace you get +10 (although this isn't called out in the log tooltip), and if you also have Elven boots or enhance ability(cat's grace) you get advantage on your roll.
If you come within 3 cells then you get detected regardless if you wait long enough, indicated by the gauge. The gauge fills up slower the better stealth you have (supposedly).
Even then I find that I occasionally get detected at greater than 3 cells with no info in the log except 'you got detected'. I'm guessing there's a missing perception check in the log.
Outside of combat, only moving stealthily: If there is no reason for the enemy to actively search for a "player", I suppose that every time the positions or circumstances change, the player does a dex(stealth) check = d20+dex and the enemy does a passive wis(perception) check = d10+wis. If the enemy's check beats the player's, he/she has been spotted.
This seems confirmed here: https://olddungeonmaster.com/2014/12/28/dd-5e-stealth-and-hiding/
But that's random and doesn't look like the gauge we see, which is more intuitive and action-oriented. Actually I'm not sure why they've implemented it like this, because it's contrary to the tabletop feeling principle they are following for this game.
In particular, if nobody moves, there is no need to do periodic checks, again this is not what we see with the gauge, and the game is in direct contradiction with the rule.
This seems confirmed here: https://rpg.stackexchange.com/questions/96256/how-often-to-roll-stealth-vs-perception
In any case, there is no trace of this process in the log, so it's very hard to tell whether they correctly use dex+10 or dex.
During combat, it's another matter, of course.
As implemented pass without trace allows your ranger to add +10 to stealth to everyone nearby, but then allows them to move as far away as they like, which is quite OP. I assume the effect should cease if a character moves more than 30ft away from the ranger.
A veil of shadows and silence radiates from you, masking you and your companions from detection. For the duration, each creature you choose within 30 feet of you (including you) has a +10 bonus to Dexterity (Stealth) checks ...
That's how I understand it too, well spotted!
Similarly should other concentration spells like shield of faith only persist while in the specified range of the caster?
It's not entirely clear in the description, is it? "A shimmering field appears and surrounds a creature of your choice within range, granting it a +2 bonus to AC for the duration". The other spell description was more precise.
Since it doesn't seem to be an instantaneous boost, but rather maintained by concentration, I'd also say the target must remain within range.