This paper considers the problem of single-server Private Computation (PC) in the presence of Side Information (SI). In this problem, there is a server that stores $K$ i.i.d. messages, and a user who has a subset of $M$ uncoded messages or a coded linear combination of them as side information, where the identities of these messages are unknown to the server. The user wants to privately compute (via downloading information from the server) a linear combination of a subset of $D$ other messages, where the identities of these messages must be kept private individually or jointly. For each setting, we define the capacity as the supremum of all achievable download rates. We characterize the capacity of both PC with coded and uncoded SI when individual privacy is required, for all $K, M, D$. Our results indicate that both settings have the same capacity. In addition, we establish a non-trivial lower bound on the capacity of PC with coded SI when joint privacy is required, for a range of parameters $K, M, D$. This lower bound is the same as the lower bound we previously established on the capacity of PC with uncoded SI when joint privacy is required.