The tagged chain dynamics in strictly two-dimensional (2D) polymer melts (where the chains are collapsed to dense spots) is considered both theoretically and by computer simulations. It is shown that the chain relaxation time in such systems scales as t_m ∝ N^α with α ≈ 1.73 (N is the number of monomer units per chain). An extended transient regime of anomalous subdiffusion is identified at t ≲ t_m where the chain centre-of-mass (CM) velocity autocorrelation function (VAF) scales as C(t) ∝ −N⁰t^−1.42. This anomalous dynamics is accounted for by the effect of the viscoelastic hydrodynamic interactions (VHI). The developed quantitative theory of the VHI-controlled chain dynamics is in good agreement, with no parameter adjustment, with the extensive simulation data. The dynamics of polymer monolayers with frictional contact to the supporting surface is considered as well. It is shown that an external (Langevin) friction γ leads to the asymptotic regime C(t) ∝ −(Nγ)^−1.37t^−0.84 that crosses over to −N⁰t^−1.42 at longer t. We also present a detailed analysis of other important factors controlling the 2D chain diffusion: finite box-size, inertial and finite compressibility effects.