Hearing aid users are challenged in listening situations with noise and especially speech-on-speech situations with two or more competing voices. Specifically, the task of attending to and segregating two competing voices is particularly hard, unlike for normal-hearing listeners, as shown in a small sub-experiment. In the main experiment, the competing voices benefit of a deep neural network (DNN) based stream segregation enhancement algorithm was tested on hearing-impaired listeners. A mixture of two voices was separated using a DNN and presented to the two ears as individual streams and tested for word score. Compared to the unseparated mixture, there was a 13%-point benefit from the separation, while attending to both voices. If only one output was selected as in a traditional target-masker scenario, a larger benefit of 37%-points was found. The results agreed well with objective metrics and show that for hearing-impaired listeners, DNNs have a large potential for improving stream segregation and speech intelligibility in difficult scenarios with two equally important targets without any prior selection of a primary target stream. An even higher benefit can be obtained if the user can select the preferred target via remote control.