I am not aware of any grand plans on how to operate public services in an anarchist society, I would inevitably have to be skeptical of any such sort of blanket approach. Many more libertarian versions of contemporary society are imaginable, but making them happen is what is lacking. Many anarchists maintain that in such a society people would voluntarily assume tasks without financial motivation to do so. There is plenty of evidence of people doing so, and writers like Kropotkin, Peter Gelderloos and Colin Ward have documented a number of examples. Also in moments of mass anti-establishment movements such as Tahrir or revolutions of the past, folks have taken even the most inglorious tasks such as trash collection upon themselves. Many people who enter service sector jobs do so, at least initially and in part, out of a motivation to help other people. I think it highly likely that if bureaucratic institutions are gotten out of the way altruism will become less stifled and undermined. It's hard to imagine a money-less society -- and far from all anarchists advocate such -- but i would like to aim for one in which services are not motived by financial considerations.
It is worth noting that this question can easily be reversed: How can public services be financed in a state-based society? More and more we are finding that the state is not willing or able to provide certain "services". The welfare state is slowly eroding, only to shore up the police state and capital state.