A private data federation enables data owners to pool their information for querying without disclosing their secret tuples to one another. Here, a client queries the union of the records of all data owners. The data owners work together to answer the query using privacy-preserving algorithms that prevent them from learning unauthorized information about the inputs of their peers. Only the client, and a federation coordinator, learn the query's output. KloakDB is a private data federation that uses trusted hardware to process SQL queries over the inputs of two or more parties. Currently private data federations compute their queries fully-obliviously, guaranteeing that no information is revealed about the sensitive inputs of a data owner to their peers by observing the query's instruction traces and memory access patterns. Oblivious querying almost always exacts multiple orders of magnitude slowdown in query runtimes compared to plaintext execution, making it impractical for many applications. KloakDB offers a semi-oblivious computing framework, $k$-anonymous query processing. We make the query's observable transcript $k$-anonymous because it is a popular standard for data release in many domains including medicine, educational research, and government data. KloakDB's queries run such that each data owner may deduce information about no fewer than $k$ individuals in the data of their peers. In addition, stakeholders set $k$, creating a novel trade-off between privacy and performance. Our results show that KloakDB enjoys speedups of up to $117$X using k-anonymous query processing over full-oblivious evaluation.