Open quantum spin systems offer a previously unexplored route to realizing deconfined quantum criticality. We consider a spin-1/2 J-Q3 chain composed of an antiferromagnetic (AFM) Heisenberg exchange and a competing multi-spin interaction favoring a valence-bond solid (VBS) state, with each spin component coupled to an independent bosonic bath. Combining non-Abelian bosonization and renormalization-group (RG) analysis with large-scale quantum Monte Carlo (QMC) simulations, we map out the phase diagram and characterize the associated phase transitions. We show that strong dissipation stabilizes an AFM phase for sub-Ohmic, Ohmic, and super-Ohmic baths. Continuous AFM-VBS transitions at finite dissipation are found upon increasing the multi-spin interaction in both the sub-Ohmic and Ohmic regimes, with their critical properties obtained through perturbative RG analysis and QMC simulations. In the Ohmic case, the critical point features spinon deconfinement and emergent O(4) symmetry. In the sub-Ohmic regime, the transition is controlled by a continuous family of nontrivial fixed points parameterized by the bath spectral exponent s and may also exhibit spinon deconfinement provided that spinons remain deconfined in the dissipative VBS phase. Furthermore, in the super-Ohmic regime, we propose a transition from the AFM to the quasi-long-range ordered (QLRO) phase in addition to the QLRO-VBS transition.