Algorithms for factorization using the decision version
All of the following algorithms either find a complete factor or a complete non-trivial factorization:


Trial division.
Pollard rho.
Pollard $p-1$ and its variants, including the elliptic curve method (ECM).
Shanks' SQUFOF.
Quadratic sieve (QS).
Number field sieve (NFS).


As far as I understand this list includes all algorithms actually used to factor integers in real life.

Finding just one bit of a factor sounds very unnatural to me, and I can't imagine an algorithm which can find one bit without finding all bits. (Even the SAT algorithm for factorization will find all bits at once.)