We theoretically consider the temporal dynamics of two coupled spin qubits (e.g., semiconductor quantum dots) driven by the interqubit spin-spin coupling. The presence of environmental noise (e.g., charge traps, nuclear spins, random magnetic impurities) is accounted for by including random magnetic field and random interqubit coupling terms in the Hamiltonian. Both Heisenberg coupling and Ising coupling between the spin qubits are considered, corresponding respectively to exchange and capacitive gates as appropriate for single spin and singlet-triplet semiconductor qubit systems, respectively. Both exchange (Heisenberg) and capacitive (Ising) coupling situations can be solved numerically exactly even in the presence of noise, leading to the key findings that (i) the steady-state return probability to the initial state remains close to unity in the presence of strong noise for many, but not all, starting spin configurations, and (ii) the return probability as a function of time is oscillatory with a characteristic noise-controlled decay toward the steady-state value. We also provide results for the magnetization dynamics of the coupled two-qubit system. Our predicted dynamics can be directly tested in the already existing semiconductor spin qubit setups providing insight into their coherent interaction dynamics. Retention of the initial state spin memory even in the presence of strong environmental noise has important implications for quantum computation using spin qubits.