In the traditional index coding problem, a server employs coding to send messages to $n$ clients within the same broadcast domain. Each client already has some messages as side information and requests a particular unknown message from the server. All clients learn the coding matrix so that they can decode and retrieve their requested data. Our starting observation is that, learning the coding matrix can pose privacy concerns: it may enable a client to infer information about the requests and side information of other clients. In this paper, we mitigate this privacy concern by allowing each client to have limited access to the coding matrix. In particular, we design coding matrices so that each client needs only to learn some of (and not all) the rows to decode her requested message. By means of two different privacy metrics, we first show that this approach indeed increases the level of privacy. Based on this, we propose the use of $k$-limited-access schemes: given an index coding scheme that employs $T$ transmissions, we create a $k$-limited-access scheme with $T_k\geq T$ transmissions, and with the property that each client needs at most $k$ transmissions to decode her message. We derive upper and lower bounds on $T_k$ for all values of $k$, and develop deterministic designs for these schemes, which are universal, i.e., independent of the coding matrix. We show that our schemes are order-optimal when either $k$ or $n$ is large. Moreover, we propose heuristics that complement the universal schemes for the case when both $n$ and $k$ are small.