X-ray and $gamma$-ray observations by the Swift satellite revealed that a fraction of tidal disruption events (TDEs) have relativistic jets. Jetted TDEs have been considered as potential sources of very high-energy cosmic-rays and neutrinos. In this work, using semi-analytical methods, we calculate neutrino spectra of X-ray bright TDEs with powerful jets and dark TDEs with possible choked jets, respectively. We estimate their neutrino fluxes and find that the non-detection would give us an upper limit on the cosmic-ray loading factor, $xi_{rm cr}lesssim20$, for Sw J1644+57. We show that X-ray bright TDEs make a sub-dominant ($lesssim5-10$%) contribution to IceCubes diffuse neutrino flux, and study possible contributions of X-ray dark TDEs given that particles are accelerated in choked jets or disk-winds. We discuss future prospects for multi-messenger searches of the brightest TDEs.