The Pliable Index CODing (PICOD) problem is a variant of the Index Coding (IC) problem, where the desired messages by the users, who are equipped with message side information, is part of the optimization. This paper studies the PICOD problem where users are subject to a privacy constraint. In particular, the following spacial class of private PICODs is investigated: 1) the side information structure is circular, and 2) each user can decode one and only one message. The first condition is a special case of the "circular-arc network topology hypergraph" class of PICOD studied in [Liu and D. Tuninetti, "Tight information theoretic converse results for some pliable index coding problems," ITW, 2018], for which an optimal solution was given without the privacy constraint. The second condition was first studied in [S. Sasi and B. S. Rajan, "On pliable index coding," arXiv:1901.05809] and was motivated by the need to keep content privacy is some distribution networks. This paper proposes both converse and achievable bounds. The proposed achievable scheme not only strictly outperforms the one in [S. Sasi and B. S. Rajan, "On pliable index coding," arXiv:1901.05809] for some values of the system parameters, but it is also information theoretically optimal in some settings. For the remaining cases, the proposed linear code is shown to require at most one more transmission than the converse bound derived by restricting the sender to only use linear codes.