Polymer-matrix composite structures with both self-monitoring and self-healing properties are proposed using a dual-material additive manufacturing technique. A double-nozzle three-dimensional (3D) printer system was adopted to fabricate these composite structures. A thermoplastic matrix with healing-agent containers was printed using one of the nozzles. The continuous carbon fibers, serving as both a sensory element and reinforcement, were embedded into the thermoplastic matrix using the other nozzle. The feasibility and effectiveness of the structures were verified through compression and three-point bending tests. Real-time self-monitoring of structural damage was performed using electrical resistance measurements. The results demonstrated that addition of 2120 Epoxy Cure agent and 2000 Epoxy Resin agent into the polymer matrix could help to repair the structural damage, with average healing efficiencies of 27.84% for non-reinforced specimens and 30.15% for fiber-reinforced specimens. In addition, digital image correlation (DIC) was utilized to analyze the reinforcing mechanism and failure behavior of the 3D printed thermoplastic composite structures reinforced with continuous carbon fibers. Integration of continuous carbon fibers and healing agents within the polymer matrix can create smart structures that can not only monitor their own structural health state, but also repair structural damage.