It depends. Smart alternators are NOT friendly to aux batteries.
Battery (surface) voltage will rise to between 14.4 and 14.7V and that's enough to charge both starter and aux battery.
A little math: a new starter battery (or a young-ish and well-treated one) will fully recharge the starting power used in a couple of minutes of normal driving. 550 amps for 5 seconds = 550/(3600/5) = 0.76Ah which should take - at, say, a nominal 50A spare alternator capacity - 0.76/(50/60) = 0.9 minutes to recharge.
Note that this is an ideal situation, it won't be all the time. The alternator won't put out 50Ah unless the engine is actually revving (say 1500rpm or more). At idle you'll have much lower alternator output and other electrical items (like 1600W subwoofers) will also consume the power and slow the charging rate down.
But because you don't have a smart alternator, your alternator will deliver as much power as it possibly could without letting up. As long as the cables connecting the main battery to the aux battery are of decent size and not too long, the aux battery will get a good charge in it.
Now, if you put your aux battery in the tub, the cable length will start to dictate the need for a DC-DC charger. If you use heavy enough cable you might get away without one, but a lead acid battery needs over 14V to charge and perferably up near 14.7V to charge faster (15V is too a little too high though). A DC-DC charger will take what voltage it can manage to get and boost it to a level that allows it to charge the battery properly.
What's really important is that the aux battery is NOT connected to the start battery while the ignition is turned off. A good DC-DC charger will achieve this because the device does the isolation itself. Whacking a heavy cable between the two doesn't allow for this unless you put a (heavy amp-rated) relay up front that's controlled by an ignition-on power source or voltage sensitive relay (VSR).