The source of an entity's funding is not relevant, but potentially relevant is whether the institution is created by law or not. Typical state universities are created by official government action, and may be specified as part of the state's constitution. Such a university is a "government university", and is subject to the limitations placed on government action (for example with respect to the 1st Amendment). There is no uniform public / private legal distinction, but the most likely grounds for making any distinction would pertain to trespassing and warrants.
For example, a military base is owned by an arm of the government and is entirely funded by the government, and individuals can be excluded from such territory even to the point of prohibiting political demonstrations – just as you can exclude an individual from your private home, including the case that they wish to conduct a political demonstration. The most relevant distinction is between a public area and a private area – it's not primarily about property ownership.
Any university can designate a particular place "private", and can also designate a place a "public forum". The library or a dining area might be private, or it might be public. A university "quad" is typically designated a public forum, and you may conduct a political demonstration there. As a public forum, permission is not required to enter, whereas a residence (dorm) is not a public forum, and entry requires permission.
Suppose that you want to have a political rally on university property, in a public forum. If the university is a private one, they may inspect your ideology, declare it to be repugnant to their interests, and prohibit the rally. If the university is a government one, the 1st Amendment prohibits them from banning your rally because your ideology is repugnant. But the right to speak freely in a public forum is not absolute: it can be regulated because of a compelling government interest. (The grounds for such regulation will be narrow, for example, there can be a requirement to schedule events and party B may be precluded from rallying at a place and time if party A has a prior claim on that place and time).
A university (government or private) can also ban an individual from being on university property. In the case of a private university, they can just do it; in the case of a government university, there has to be "due process". So, what a university is allowed to do depends on the particular action, and the most relevant distinction is between government vs. private universities, with source of funding being entirely irrelevant.