The ring structure of certain polymers in nature, like proteins and DNA indicates a benefit compared with the linear form. Transcriptional regulation in higher eukaryotes is maintained among others by the formation of chromatin loops. Experimental studies revealed that different chromosomes as well as chromatin regions on one single chromosome tend to be segregated into distinct territories. Here we study a system of two rings in both catenane and bonded topology as a toy model for the influence of loops and topological constraints on the polymers conformational properties. Athermal Monte Carlo simulations reveal that the mean square radius of gyration (R-gyr(2)) of catenated or bonded rings follows a similar scaling exponent as isolated rings, which in turn is close to the value of nu approximate to 0.588 for a self-avoiding walk. However, the effective segment length is larger for catenated rings, reflecting the swelling of the polymers. The shape of catenated and bonded rings, in contrast, shows pronounced differences even in the limit of infinite chain length. We observe a strong tendency toward segregation for the bonded topology in comparison with a similar ring-linear and linear-linear system. The orientation of the rings' gyration ellipsoids is slightly perpendicular, trying to minimize the overlap area. These findings indicate that loops might play an important role in the entropy-driven segregation of chromatin.