I guess the XB1 running noise, with its large fan, just has to be more quiet than the PS4's. But maybe they suck at engineering even more than what we already saw so far.
However I don't think either fan noise will be a problem. Neither of the current gen consoles have really loud fans. The original 360 maybe. The fat PS3 makes also quite a racket on the highest fan setting, but unless you turned the volume off of the game, it's not really distracting. This is of course subjective...
During movies, this was actually more noticeable, but then again the fan is on lower modes anyway. This should get even better on the next consoles, considering both consoles should not get hot at all while playing movies. Stepless fans on PS4 will also improve on that.
In a complete silent room, I can hear when my slim PS3 is on while it downloads stuff for PS+ etc., but even that is not bad.
I would assume that in the low power mode of the PS4, the fan is even completely off.
The loudest part will anyway be the disc drive during streaming/installing, this will be same for both XB1 and PS4, assuming they have the same max speeds. XB1 will of course be a much bigger improvement on 360 in this regard, the high spinning DVD reads of the 360 were/are always the most loud part.
Btw, it would be neat if the PS4 allocates e.g. 50 gb of the HDD for the current game in play, unless you choose to permanently save the game data for that game on the HDD. So it will install while you play, every time you switch to a different game of course, but at the BD drive speeds it should only be like during the first 10 minutes (and 6x is not really loud anyway). That way you would never have to see installs nor worry about making room on the HDD for them.