Integrated photonics represents a technology that could greatly improve quantum communication networks in terms of cost, size, scaling, and robustness. A key benchmark for this is to demonstrate their performance in complex quantum networking protocols, such as entanglement swapping between truly independent photon-pair sources. Here, using two independent, asynchronously-pumped, integrated Si3N4 microring resonator photon-pair sources, operating in the continuous-wave regime with time-resolved detections, we obtained state of the art Hong-Ou-Mandel (93.2 +/- 1.6%) and entanglement swapping (91.2 +/- 3.4%) visibilities, while maintaining high rates. The time-resolved detection facilitates high spectral purities without the need for spectral filtering. Our results demonstrate the potential of such telecom-band sources for practical, real-world quantum communication.