I can hear pitch very well. So when something is out of tune it normally stands out easily to me. I hate watching the talent shows on TV for that reason.... singers often are OOT.
In the studio, I have several tuners. Since there can often be a slight discrepancy of a few cents between tuners, I tend to use one tuner all the time. I also checked it before using it by playing A-440 on my midi keyboard and making sure it is properly calibrated to that pitch.
If I'm simply rehearsing a part or jamming to figure out a part, I'm not so worried about accurate tuning. Close is OK for that as long as it doesn't make me cringe.
When recording.... it has to be tuned and I will tune again as often as is necessary during the session.
Intonation on the guitar neck: If the intonation is off, it should be adjusted properly. Having said that..... I played with a guitarist who was constantly playing out of tune.....and blamed it on his guitar. ( a real Fender Strat) He mentioned that my SG always played in tune..... so we swapped guitars for one set one night.
Yep... you guessed it. the Strat was perfectly in tune that whole set for me and the SG was horribly OOT for him. It turned out to be our playing technique rather than the intonation of the guitars. The guitars were both set up properly for intonation.