Short answer. Both, if the ditch is common — and a ditch between two estates is presumed common unless a title or a sign shows otherwise. The decisive sign is where the spoil lies: if the earth dug out sits on one side only, that side owns the ditch alone and maintains it alone.
What the law says
Ditches or drains opened between two estates are also presumed as common to both, if there is no title or sign showing the contrary.
Civil Code, Article 661 — Ditches and Drains Between Estates. Read the full provision →
Common ownership means a shared burden
The starting position is that a ditch running between two properties belongs to both. That is what decides who cleans and repairs it: an owner in common carries his share of keeping the common thing serviceable, and the neighbour who has been clearing the channel alone for years has been subsidising the other rather than acquiring anything by it. The presumption applies to ditches or drains opened between two estates, so a channel that lies wholly inside one lot is not reached by it at all, whatever water happens to flow through.
The spoil test
The article then gives a test anyone can apply on the ground. There is a sign contrary to part-ownership whenever the earth or dirt removed to open the ditch or to clean it is only on one side thereof, and where that is so, the ownership of the ditch shall belong exclusively to the owner of the land having this exterior sign in its favor. The reasoning is simple: the person who dug it and keeps digging it out leaves the spoil on his own side. The bank of accumulated earth is a record of who did the work, written on the land itself.
What the test does not settle
Read the sign carefully before relying on it. The article speaks of earth on one side only, so spoil heaped along both banks, however unevenly, leaves the presumption of commonness standing. And the sign proves ownership of the ditch, not a right to do as one likes with it — an exclusive owner still may not use the channel in a way that damages his neighbour's land. Nor does the presumption apply where a title exists, so a deed, an annotation, or an approved plan showing the ditch will be read before anyone looks at the dirt.
Recording the position before it changes
The evidence in this class of dispute is perishable. Spoil banks are levelled, ditches are concreted over, and the exterior sign that would have settled the question disappears in an afternoon's work. So photograph the ditch and both banks, dated, before any clearing is done, and get a relocation survey showing where the channel sits in relation to the boundary. Then check both certificates of title for annotations. If maintenance is going to be shared going forward, an agreement in writing about who does what, and how often, is worth far more than the custom nobody wrote down.