Is there an efficient algorithm that allows to construct a cryptographically secure $n$-bit hash function based on two unkeyed $n$-bit permutations?
It is impossible to construct a collision-resistant hash function from one-way permutations, via a black-box construction. Thus, the answer in principle is no; such a construction is impossible. Finding collisions on a one-way street: Can secure hash functions be based on general assumptions? by Simon (Eurocrypt'98). This answer does not relate to heuristic constructions, but you cannot do this with a proof of security (unless it's not black box).

The question here actually relates to "ideal" permutations, and under such an assumption it may be different. (Although, I'm not sure what is meant by "public" ideal.) My answer referred to (non-ideal) one-way permutations.