  While the radiation mechanism of fast radio bursts (FRBs) is unknown,
coherent curvature radiation and synchrotron maser are promising candidates. We
find that both radiation mechanisms work for a neutron star (NS) central engine
with $B\gtrsim 10^{12}$ G, while for the synchrotron maser, the central engine
can also be an accreting black hole (BH) with $B\gtrsim 10^{12}$ G and a white
dwarf (WD) with $B\sim 10^8-10^9$ G. We study the electromagnetic counterparts
associated with such central engines, i.e., nebulae for repeating FRBs and
afterglows for non-repeating FRBs. In general, the energy spectrum and flux
density of the counterpart depend strongly on its size and total injected
energy. We apply the calculation to the nebula of FRB 121102 and find that the
persistent radio counterpart requires the average energy injection rate into
the nebula to be between $2.7\times10^{39}~{\rm erg/s}$ and
$1.5\times10^{44}~{\rm erg/s}$, and the minimum injected energy be
$6.0\times10^{47}~{\rm erg}$ in around $7$ yr. Consequently, we find that for
FRB 121102 and its nebula: (1) WD and accretion BH central engines are
disfavored; (2) a rotation-powered NS central engine works when
$1.2\times10^{12}~{\rm G}\lesssim B\lesssim 7.8\times10^{14}~{\rm G}$ with
initial period $P<180$ ms, but the radio emission must be more efficient than
that in typical giant pulses of radio pulsars; and (3) a magnetic-powered NS
central engine works when its internal magnetic field $B\gtrsim 10^{16}$ G. We
also find that the radio-emitting electrons in the nebula could produce a
significant rotation measure (RM), but cannot account for the entire observed
RM of FRB 121102.
