Your generator set will only charge its own starting battery (and as Genesis pointed out, it should have its own starting battery) and supply current to the AC mains when it is running. However, if your battery charger is turned on, if (for example) you are at anchor, with your main engines off and your generator set running, with the battery charger turned on, it WILL charge all your batteries while it is running.
Most of our boats were wired by Hatteras in a fashion similar to what Genesis describes; the starting loads are split between the two banks, and in some boats the house loads are also split between the two banks. This has the potential to leave you with a boat that won't start on either engine, plus nothing to run your house loads. This is one of the reasons that a dedicated generator starting battery is desirable; if everything else is discharged to the point of uselessness, the generator can recharge your starting batteries enough to get you moving again.
Modern boats, especially trawlers, from which we could often learn something, are wired differently: there is a dedicated starting bank of starting-only batteries, used only to start the engines, and one engine's alternator charges that bank only. There is a dedicated house bank, of deep cycle batteries, which is big enough that it can also start the engines through a relay (which most Hatteras yachts already have), and that bank is charged by the other engine's alternator, usually using a smart regulator. Finally, there is a combiner, which will distribute and balance out the charging, also ensuring that if one bank is totally down it does not discharge the other bank. Also useful is a battery monitor which will let you keep track of the amount of charge left in each bank.
I had my boat rewired this way a few years ago and it has worked well so far. One of the biggest changes we made was using better voltage regulators, which make faster and better use of the alternators' output- so-called smart regulators. (what they are is multi-stage regulators with adjustable charging profiles) Most boats don't need bigger alternators, but they DO need better regulators. That made a big difference, too.