Do these amps produce an AC circuit with $+$ and $-$ in phase with their outputs or is their gain like a shift?

I'm not sure I read or understood your question correctly... but the amps can indeed result in phase shifts, i.e., a difference in phase between the input and output sinusoids. The phase shift can be intrinsic to the amplifier operation. For example, for the CE amp, the output is inverted with respect to the input, which represents a half-cycle or $180^\circ$ phase shift. There can in addition be phase shifts associated with ``roll-off'', if the frequency is such that there is capacitive-filter attenuation.