Who pays the costs to maintain an easement?

The question you pose begs the question whether the road is a truly "private" road, and not a public road for which the city or county within which the road lies would be responsible for repairs. Assuming that the road is private, the first step is to determine by a title search to determine if any right-of-way easment has been granted for any use of that road/driveway. If that is the case, all the owners that "use" the road have obligations to maintain it under Civil Code section 845.
Absent a written agreement, recorded and running with the land (a term of art that a real estate attorney can explain to you) each party's obligations are theoretically determined by the burden or "use" to which they put the road. Figuring out what contribution is required is where the difficulty commences.
First, the obligation to "maintain," does not mean to improve or make the road/driveway pretty, only to bring the road back to its original condition of utility. As you might expect, nobody wants to pay voluntarily to repair a road, and conflicts between owners are frequent and heated. The way the statute is written creates a sliding scale of burdens upon the owners depending on the degree of use, about which reasonable minds can differ. Frequently, the parties are completely unreasonable and nothing gets done informally, either to get contributions or to create a road maintenance agreement. Then the parties may resort to the courts, and arbitration, at their own expense, to have the respective uses and contributions for repairs determined by an arbitrator.
I infer from our question that the owners of properties on your private road have not been able to agree, which is common, and a direct result of the unwilingness of this and other developers to go to the expense to put the road under the municipality's jurisdiciton before you came into ownership and the archaic statue governing private roads.
Your first stop is to check your title insurance to determine if an easement has been created by a recording. If not, you may be able to seek to have an easement estatblished by agreement or implication, again, at great expense.
Consult an attorney in Riverside that is conversant in easements for rights of way and have him or her advise you of your rights and the rocky road you may face getting repairs done. I sympathize with you, as I have endured the same difficulties you will face in providing maintenance of a private road to my own home.
Good luck.