Public property is our neighbor's property as well as ours. Seems to me that the logical extension of your statement is that: "You and I have no right to dictate what my neighbor does on his personal public property, so long as it does not violate the rights of others."
"It is acceptable…to define what one can and cannot do on public property, because we collectively own the property."
But on what basis is it acceptable? Collective ownership can't be the basis because the "one" in question is one of the collective owners. You already know the answer. It's in the logical extension of your statement that I noted above: a violation of the rights of other.
So, in response to your assertion that "The government has the right, and the duty, to arrest you if you attempt to drive on a public road, while under the influence of a drug, while impaired visually or mentally, or even if you have not had sufficient sleep to stay awake, while driving": In what way does driving while impaired violate the rights of others such that it forms the basis for the Government's right and duty to arrest one if one attempts to drive on a public toad while impaired? In other words, why does the government have this right and duty if there is no direct harm or violation of rights?
"If we (you, I, the drunk guy, and all other citizens) deem drunk driving to be forbidden on our property, then how are his rights violated?"
If we all agree, then his rights are not violated because we have all agreed to give up some of our individual rights for whatever reason. And that is an answer to the question "Why does the government have this right and duty if there is no direct harm or violation of rights?"
"I do not see how we violate anybody's rights, when we deem drunk driving on public roads - our roads - to be illegal and punishable at the hands of government."
Collective agreements aside for the moment, we are restricting someone's liberty because of harm they might or might not cause, not because they actually hurt anybody else or violated someone else's rights. It's like prohibiting free speech to someone because they might slander someone else, even though they also might not.
By the way, your argument:
Implies that if private property is collectively owned, one of the collective owners can be denied its use by the others for reasons that they find acceptable.
Ignores the rights of the minority. (Remember that one of the basics of our system is the protection of the rights of the minority from the whims of the majority.)
Implies that one who is speech impaired could be denied "free speech" if the majority found that to be acceptable.