Amplitude compression is an indispensable feature of contemporary audio production and especially relevant in modern hearing aids. The cortical fate of amplitude-compressed speech signals is not well studied, however, and may yield undesired side effects: We hypothesize that compressing the amplitude envelope of continuous speech reduces neural tracking. Yet, leveraging such a "compression side effect" on unwanted, distracting sounds could potentially support attentive listening if effectively reducing their neural tracking. In this study, we examined 24 young normal hearing (NH) individuals, 19 older hearing-impaired (HI) individuals, and 12 older normal hearing individuals. Participants were instructed to focus on one of two competing talkers while ignoring the other. Envelope compression (1:8 ratio, loudness-matched) was applied to one or both streams containing short speech repeats. Electroencephalography allowed us to quantify the cortical response function and degree of speech tracking. With compression applied to the attended target stream, HI participants showed reduced behavioral accuracy, and compressed speech yielded generally lowered metrics of neural tracking. Importantly, we found that compressing the ignored stream resulted in a stronger neural representation of the uncompressed target speech. Our results imply that intelligent compression algorithms, with variable compression ratios applied to separated sources, could help individuals with hearing loss suppress distraction in complex multitalker environments.