In this paper, we study the impact of pilot contamination in a system where two operators serve their respective users with the assistance of two wide-band reconfigurable intelligent surfaces (RIS), each belonging to a single operator. We consider one active user per operator and they use disjoint narrow frequency bands. Although each RIS is dedicated to a single operator, both users' transmissions are reflected by both RISs. We show that this creates a new kind of pilot contamination effect when pilots are transmitted simultaneously. Since combating inter-operator pilot contamination in RIS-assisted networks would require long pilot signal sequences to maintain orthogonality among the users of different operators, we propose the orthogonal configurations of the RISs. Numerical results show that this approach completely eliminates pilot contamination, and significantly improves the performance in terms of channel estimation and equalization by removing the channel estimation bias.