The answer depends on which system your person is in, but in most cases the information is publicly available and accessible without making a single phone call.
For federal inmates, the Bureau of Prisons inmate locator at bop.gov is the most reliable starting point. Search by name or federal register number and the record will show the facility, the sentence information, and the projected release date as currently calculated by the BOP.
For state inmates, every state department of corrections maintains its own offender search tool on their official website. Search the state name plus department of corrections offender search and you will find it. Results vary by state in how much detail they show, but most include the current facility and a projected release date.
For county jail inmates, the picture is less consistent. Some counties publish online inmate rosters with release date information. Many do not. For those that do not, VINELink at vinelink.com is the best alternative. It pulls custody data from participating facilities across the country and can surface current status even when the facility itself does not have a public-facing locator.
InmateAid's inmate locator is another resource worth trying across all three categories. If the locator comes up empty or shows outdated information, email aid@inmateaid.com with the person's name and last known location and the team can help track down the current record.
A few things worth knowing about release dates once you find them. The date shown is a projected date based on the current sentence calculation including good time credits. It can change if the inmate picks up disciplinary infractions or earns additional program credits. Treat it as the most accurate current estimate rather than a guaranteed date.