Religious handbills should be permitted to be distributed on public ground under the constitutional right to freedom of speech. This is also a case of freedom of religion so long as that religion is not oppressing the rights of others in any way. Also, this is not a case involving the separation of church and state.
The first amendment supports free speech. By not allowing the distribution of religious handbills on public grounds, we are not supporting their first amendment rights. There is no reason why this group, just because they are religious, should be exempt from the respect we show others. They should be freely allowed to distribute their handbills.
Yes, the distribution of religious handbills on public grounds should be permitted, because that is freedom of speech. That is not the government sanctioning any particular religion; rather, they are just allowing the people to exercise their own. That is the difference between freedom of religion and freedom from religion.
Public grounds are, basically, in service of the public and therefore should not be subject to the whims of certain groups without reason. Unless the group distributing religious handbills are truly infringing on others, the behavior should be permitted as an exercise in free speech and freedom of religion. The oft-touted "freedom from religion" would in this case infringe on the aforementioned rights.
They should not be allowed to be distributed on public grounds. Our country has a strict separation between church and state. Allowing handbills like this would give the impression of an endorsement by the state of one religion over another, alienating the public. Public grounds take public money to maintain, so it would also be unfair to allow one religion to take advantage of it.