Are one side signals affected by other side signals on multilayer PCBs?
Yes, different conductors on a PCB have some capacitance between them.  This can cause coupling of signals between traces.  The usual ways to avoid this are:

Distance.  Capacitance between traces falls off with distance.

Shielding.  Put another low-impedance conductors (like ground) between the two traces.  They will each couple to this center conductor, but not to each other.  This applies to parallel traces in the same layer, and something like a ground or power plane on a layer between the two conductors.

Good design.  Try to limit voltage dV/dt, since capacitive coupling is proportional to it.  Try to reduce the impedance of traces you don't want picking up signals from elsewhere.  Feedback around a amplifier section, for example, also helps to attenuate noise picked up within that section.



You also have to consider inductive coupling.  Passing current thru a wire causes a circular magnetic field around the wire.  This also works in reverse in that a changing circular magnetic field around a wire induces a voltage along that wire.

when two conductors are parallel and near each other, each is partially within the magnetic field of the other.  Running a alternating current thru one will induce some alternating voltage in the other.  If you think this is far-fetched, consider that transformers work by deliberately exploiting this principle.

Avoiding magnetic coupling can get tricky, because something like a ground plane doesn't block it.  In general, avoid long runs of parallel conductors where one carries high current and the other is sensitive to induced noise.  Just like with capacitance, distance helps.  For magnetic coupling, orientation also matters.  Two conductors crossing at right angles in theory have no magnetic coupling between them.