IGR J17591$-$2342 is a new accreting millisecond X-ray pulsar (AMXP) that was recently discovered in outburst in 2018. Early observations revealed that the sources radio emission is brighter than that of any other known neutron star low-mass X-ray binary (NS-LMXB) at comparable X-ray luminosity, and assuming its likely $gtrsim 6$ kpc distance. It is comparably radio bright to black hole LMXBs at similar X-ray luminosities. In this work, we present the results of our extensive radio and X-ray monitoring campaign of the 2018 outburst of IGR J17591$-$2342. In total we collected 10 quasi-simultaneous radio (VLA, ATCA) and X-ray (Swift-XRT) observations, which make IGR J17591$-$2342 one of the best-sampled NS-LMXBs. We use these to fit a power-law correlation index $beta = 0.37^{+0.42}_{-0.40}$ between observed radio and X-ray luminosities ( $L_mathrm{R}propto L_mathrm{X}^{beta}$). However, our monitoring revealed a large scatter in IGR J17591$-$2342s radio luminosity (at a similar X-ray luminosity, $L_mathrm{X} sim 10^{36}$ erg s$^{-1}$, and spectral state), with $L_mathrm{R} sim 4 times 10^{29}$ erg s$^{-1}$ during the first three reported observations, and up to a factor of 4 lower $L_mathrm{R}$ during later radio observations. Nonetheless, the average radio luminosity of IGR J17591$-$2342 is still one of the highest among NS-LMXBs, and we discuss possible reasons for the wide range of radio luminosities observed in such systems during outburst. We found no evidence for radio pulsations from IGR J17591$-$2342 in our Green Bank Telescope observations performed shortly after the source returned to quiescence. Nonetheless, we cannot rule out that IGR J17591$-$2342 becomes a radio millisecond pulsar during quiescence.
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