What does an oscillating magnetic field produce? An alternating current produces a magnetic field.  A oscillating charge produces light, which is an oscillating electric, and magnetic field mutually perpendicular to each other, and their motion is the Poynting Vector .  What can a oscillating magnetic field produce?
 A: As you point out, an oscillating electric charge (and by that I assume you mean a charge moving back and forth) generates both an oscillating electric field but also an oscillating magnetic field. This is straight forward to understand from the fact that charge in motion produces a magnetic field. This is also built in to Maxwell's equations (that describe electric and magnetic fields), that an oscillating electric field generates a magnetic field. Similarly, if you create a magnetic field that oscillates you automatically create an oscillating magnetic field. Indeed, the link between electricity and magnetism was originally unsuspected until Oersted discovered there was some relation between them.
A: You're simply asking about Faraday's observations and law: a time varying magnetic field "produces" an electric field and this production is described precisely by Faraday's law:
$$\nabla\times \vec{E} = -\partial_t\,\vec{B}$$
It may be more familiar to you in its integral (but altogether equivalent) form:
$$\oint_\Gamma \vec{E}\cdot\mathrm{d}\vec{r} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d} t}$$
which says that the voltage around a closed loop $\Gamma$ is equal to the rate of change of magnetic flux through the loop.
So, if you like, a time varying magnetic field "produces" an electromotive force (voltage) around any loop it threads.
Note that I put the word "produces" in quotes as "produces" ordinarily implies a causal link between that which produces as the cause, and that which is produced as the effect. But Maxwell's equations alone are wholly acausal; if you invert the direction of time in any valid solution to them, you get an equally valid solution. So it is often not really appropriate (and often downright confusing) to speak of them and their solutions in cause-effect language. Moreover, electric and magnetic fields are not separate entities but different components of the same, Faraday tensor, so that the electric and magnetic fields arising from a source depend on the motion of an observer of that source.
