IANAL - For the GPL/LGPL, at least, the license is triggered upon distribution, so you are not required to do anything if you're using it on the server side of a website. Common courtesy, of course, would say if you make any bug fixes, etc., that you should contribute that back to the community. In fact, it would be nice of you to redistribute it just to be nice (I understand how this could be impossible if you mix non-GPL compatible code with GPL code though).
Note that this is why the AGPL was created. If you find something that's AGPL, you can't use to run a public website without releasing any changes.
The only material difference between the GPL and the AGPL is that the AGPL extends the concept of distribution to include "providing the web interface of a program to an internet-facing customer."
If you really want a workaround, the way you do it is by wrapping the AGPL library in an executable, and calling it from the command line. The FSF calls this "arms-length communication;" it means that your AGPL library is not "linked" with your executable via early binding or late binding programming language mechanisms.
See here, where it says:
In many cases you can distribute the GPL-covered software alongside
your proprietary system. To do this validly, you must make sure that
the free and non-free programs communicate at arms length, that they
are not combined in a way that would make them effectively a single
program.
The difference between this and “incorporating” the GPL-covered
software is partly a matter of substance and partly form. The
substantive part is this: if the two programs are combined so that
they become effectively two parts of one program, then you can't treat
them as two separate programs. So the GPL has to cover the whole
thing.
If the two programs remain well separated, like the compiler and the
kernel, or like an editor and a shell, then you can treat them as two
separate programs—but you have to do it properly. The issue is simply
one of form: how you describe what you are doing.
In short, the AGPL library has to be a standalone program, and your application has to be able to function without it.
As always, if you have real legal concerns, you should consult a real lawyer who specializes in this sort of thing.
Best Answer
Yes, you are obliged to make the source code available, under the same terms as the AGPL - which are almost equivalent to the GPL, by design.
Technically, the GPL requires that source is distributed only to people who ask for it, and who have received a copy of the binary. You are not obliged to go beyond that, although you cant stop, eg, me from giving the binary to Mary, and Mary asking for the source through me.
The additional nature of the AGPL is that if you put the AGPL component into a service that is never distributed, but is used by others, you are also obliged to make it available to them.
Since the AGPLv3 and the GPLv3 are almost identical, "Does the GPL allow me to sell copies of the program for money?" applies to you also. In other words, "yes, totally, go right ahead".
You can't stop me buying a copy, getting the source, and distributing both free of charge - but you can totally put it, eg, in the iOS market and charge anything you want. (You can't stop me doing the same, and pricing it one cent lower than you do, though, so the model can be ... risky.)