We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I = a, b | 2a = 2b = 0, a 2 = b, ab = 0 and the noncommutative ring E = a, b | 2a = 2b = 0, a 2 = a, b 2 = b, ab = a, ba = b. We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.