The operations of a road association are often compared to the operations of a town. So I think the answer would be to use Title 23 section 705 as a model. Here's a link: https://legislature.maine.gov/legis/statutes/23/title23sec705.html
Basically what it says is that if a person wants to install a new driveway to an existing road, that person is responsible for installing the culvert. But if a town wants to put a ditch across an existing driveway, it's the town's responsibility to install and maintain the culvert. So in your case, I believe the road association should pay for the culvert installation.
You also want to bear in mind 23 MRS 3101 paragraph 5-A:
5-A. Easements. A road association under this subchapter may negotiate an easement for the installation of a ditch, drain, culvert or other storm water management infrastructure to benefit the private road, private way or bridge. The easement must specify when a ditch, drain, culvert or other storm water management infrastructure must be maintained and include reasonable performance standards to guide the timing and extent of its upkeep and repair. The easement must also be recorded at the registry of deeds in the county in which the property subject to the easement is located. A ditch, drain, culvert or other storm water management infrastructure subject to an easement under this subsection must be under the control of and maintained by the road association.
I believe that if your ditch is within the bounds of the right of way of the road, you shouldn't need to get an easement to install the ditch. But if you need to go beyond the bounds of the right of way, you should be getting easements from the landowners. If your road is in a subdivision, you should be able to find the specified width of the right of way on the subdivision plan. The deeds of the abutters probably give reference to the book and page number of the subdivision plan, and you can then look it up online at the Registry of Deeds website for your county, here:
https://www.maineregistryofdeeds.com/
If you have difficulty finding the width of the right of way, let me know and I'll see if I can help.