Will there be any interference between the signals? Is this going to
cause any problems?
It's going to cause problems for sure. Think about this.
A single op-amp with negative feedback seeks to maintain the differential voltage across its inputs at zero volts by adjusting its output voltage. If the op-amp has got an input offset voltage error of +1 mV then the op-amp will have an error on the output. That error is generally gain*input_offset. So if the gain is -1 (as per an inverting op-amp) there will be a -1mV error at the output.
This is normal and accepted by design engineers however, if you have two op-amps (one with +1 mV input offset error and the other with -1 mV input offset error) they will compete to set the common inverting input at a voltage that suits them. There is no winner here; one op-amp will output against a power rail and the other may do the same but the opposite power rail.