Inside the facility, partner notification is complicated by the fact that sexual activity between inmates is officially prohibited, which means acknowledging who a person's sexual partners are inside requires admitting to conduct that can result in disciplinary action. In practice, prison health services may attempt contact tracing, particularly for serious infections like HIV, but the process is inconsistent and often limited by the same bureaucratic and privacy constraints that apply in any healthcare setting.
For partners on the outside, the situation depends on what the STD is and what state he is in. Public health law in most states requires healthcare providers, including those at correctional facilities, to report certain communicable diseases to the local or state health department. For conditions like HIV, gonorrhea, chlamydia, and syphilis, health departments have partner notification programs and may reach out to known partners if contact information is available.
Here is the more important point: if you are sexually active with someone who is or has been incarcerated, getting tested yourself is the responsible move, regardless of whether you expect to be formally notified. Prison health services are not a reliable notification system, and relying on the facility to tell you something you should know about your own health is not a position worth being in. Get tested directly, and do so regularly if the relationship is ongoing.