In this work, we investigate two secure source coding models, a Helper problem and a Gray-Wyner problem. In both settings, the encoder is connected to each of the legitimate receivers through a public link as well as a private link; and an external eavesdropper intercepts every information that is sent on the public link. Specifically, in the Helper problem, a memoryless source pair (S0; S1) is to be compressed and sent on both links such that the component S0 can be recovered losslessly at the legitimate receiver while being kept completely secret from an eavesdropper that overhears on the public link, and the component S1 is recovered lossily, to within some prescribed distortion level, at the legitimate receiver. In the Gray-Wyner model, a memoryless source triple (S0; S1; S2) is to be compressed and sent to two legitimate receivers, such that the component S0 is recovered at both receivers losslessly and kept secret at an external eavesdropper that listens on the public link; and the component Sj, is to be recovered lossily at Receiver j, j = 1, 2. We establish single-letter characterizations of the optimal secure rate-distortion regions of both models. The analysis sheds important light on the role of the private link(s), i.e., for the transmission of the source S0 or for sharing a secret key that is then used to encrypt the source S0 over the public link.