One way to achieve zero-knowledge is to simply compose Jolt with a zero-knowledge SNARK like Groth16. That is, use Groth16 (which is ZK) to prove that one knows a Jolt proof. Composing with Groth16 or Plonk is how most zkVMs get low on-chain verification costs anyway, and it also "adds" ZK. This approach is on Jolt's roadmap, although it will take some time to complete (as it requires representing the Jolt verifier in R1CS or Plonkish, which is a pain).
There are also ways to make Jolt zero-knowledge without invoking SNARK composition. For example, rendering sum-check-based SNARKs zero-knowledge without using composition was exactly the motivation for Zeromorph, which introduces a very efficienct zero-knowledge variant of KZG commitments for multilinear polynomials.
If we use the Zeromorph polynomial commitment scheme, the commitment and any evaluation proof are hiding (they reveal nothing about the committed polynomial, and still give the verifier a commitment to the requested evaluation of the committed polynomial). One still needs to ensure that the various applications of the sum-check protocol in Jolt also do not leak any information about the witness. Here, techniques based on masking polynomials apply (see Section 13.3 of Proofs, Arguments, and Zero-Knowledge for a sketchy overview). However, the use of masking polynomials requires the prover to be able to commit to non-multilinear polynomials and hence introduce significant (but surmountable) issues.
A similar approach to achieving ZK also applies when using a hashing-based polynomial commitment scheme like Brakedown. Roughly, to randomize all values sent by the prover during the Brakedown evaluation proof, it suffices to pad the committed polynomial with sufficiently many random values. One does need to lightly modify the Jolt polynomial IOP to "ignore" these extra, random committed values.
A final technique to render all of the sum-checks ZK without SNARK composition is given in Hyrax (based on old work of Cramar and Damgard). Roughly, rather than the prover sending field elements "in the clear", it instead sends (blinded, hence hiding) Pedersen commitments to these field elements. And the verifier exploits homomorphism properties to confirm that the committed field elements would have passed all of the sum-check verifier's checks. See Section 13.2 of Proofs, Arguments, and Zero-Knowledge for additional discussion.