There are claims that the killing is mutual. There are claims that it's one sided.
Which one is right?
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While @nic provides the crux of the answer, I thought I would elaborate a bit.
The immediate event that triggered the mass killing was the kidnapping and murder of 6 generals in an apparent coup d'etat attempt against Sukarno by the so-called 30 September Movement. It is not known whether and to what extent the communist PKI (Communist Party of Indonesia) was involved in this killing, but the "official" story under Suharto's subsequent regime was that the PKI was to blame. The scale of this single event pales in comparison to the period of mass killing that followed, but I thought it relevant to mention.
The question refers to "killing between communists and Muslims" but this is a misleading description of what happened, which reflects the "official" version of events. Especially in the beginning but to some degree throughout the massacres of suspected PKI supporters was led by the army. Particularly in this phase and in central Java where the communists were most well organized, PKI supporters did fight back to some extent but were quickly overwhelmed. (source)
Muslim groups (specifically santri, as many of the victims were also Muslim, but typically abangan) did quickly become important participants in areas where they were the majority. In Bali, there were Hindu groups that played a similar role. While religious groups did a lot of the "dirty work" in perpetrating the killing, it is clear that they did not initiate it in the country as a whole. It was possible for such groups to participate in the killing only because those communists who were in any position to defend themselves were defeated by the army. For more on the specific role of the army in the mass killing, here is a book The Army and the Indonesian Genocide: Mechanics of Mass Murder by Jess Melvin.
The most significant case of reprisal killing I have seen mention of came in 1968 near Blitar, when about 60 members of Nahdlatul Ulama were killed. This was a santri Muslim group which participated in the killings of 1965. This was part of a more general armed insurgency as described in this article. While this was certainly a significant event, I would still say that the answer to the headline question is "no", the killing in 1965 was not two-sided.
The Wikipedia article says:
Although some PKI branches organised resistance and reprisal killings, most went passively to their deaths
Citing this reference I can not access:
McDonald (1980), p. 53; Friend (2003), p. 115.
McDonald, Hamish (1980). Suharto's Indonesia. Melbourne: Fontana Books. ISBN 0-00-635721-0.
Friend, T. (2003). Indonesian Destinies. Harvard University Press. ISBN 0-674-01137-6.