This is interesting; the couple bought the street which has a privacy gate - do they own the gate too? Is is possible that they owned the street but could not legally access it (by not owning the gate)? Does that affect the case/transaction? I'm not familiar enough with the details.
I don't what the specific details are in this case, but there may be an easement. Generally property owners are entitled to access their real property. For example, if an owner purchases a plot of land that has no access except by crossing another property there is an "easement by necessity" for the former owner to cross the property of the latter to access his property.
Whether the sidewalk owners can enforce it is another matter. (See also Kosla's jackassery around denying water access.)
Whether the sidewalk owners can enforce it is another matter. (See also Kosla's jackassery around denying water access.)