Segregating important stimuli from distractors is crucial for successful speech perception. Neural activity aligned to speech, also termed “neural entrainment”, is thought to be instrumental for this purpose. However, the relative contribution of neural entrainment to target and distractors in speech perception remained unclear. In this study, we used transcranial alternating current stimulation (tACS) to manipulate entrainment to two simultaneously presented sequences of rhythmic speech while participants attended to one of them. A random temporal relationship between speech streams allowed us to disentangle effects of tACS on target and distractor processing, and to examine their combined effect on a behavioural measure of speech perception. We found that the phase relation between tACS and both target and distracting speech modulated word report accuracy, and to a similar degree. The phasic modulation of target processing correlated with that of the distractor across subjects, and their combined effect on speech perception was stronger than each of the two alone. In contrast to our expectation, the tACS phases leading to most accurate perception were uncorrelated between target and distracting speech modulations. Together, our results suggest that entrainment to target and distracting speech jointly and causally contributes to speech perception. They also demonstrate how effect sizes might be increased in future work and for technological or clinical applications. Significance statement Synchronising neural activity to moments of important information in speech is beneficial for its perception. Successful speech perception is a consequence of both target processing and the suppression of potential distractors; so far, it has remained unclear whether brain-speech synchronisation plays a more important role for either of these two tasks. We used transcranial brain stimulation (tACS) in a multi-speaker scenario to independently manipulate synchronisation to target- or distracting speech, respectively. We found that both manipulations led to changes in speech perception that were of comparable magnitude, and their combined effect was stronger than either of them alone. These results give an insight into the cortical processing of competing speech streams and how brain-speech synchronisation causally contributes to their perception. Our findings also have implications for practical applicability of brain stimulation protocols, showing how they could be used and made more effective in improving speech perception. This approach could have potential for medical application such as additional support in hearing aids.