It depends entirely on the facility and the inmate's custody level. Many prisons and jails do offer contact visits where you sit in the same room as the inmate without any barrier between you. Others use non-contact visitation where you speak through glass or via phone handsets with no physical proximity. Some facilities have moved to video-only visitation, which means you are not in the same room at all.
State prisons at lower security levels, federal camps, and minimum-security facilities are more likely to have open room contact visits. County jails and higher-security state facilities more often use the glass and phone setup. Maximum-security units frequently restrict visitation to non-contact or video only.
The only reliable way to know what applies at a specific facility is to call and ask. Request to speak with the visiting room officer or the visitation coordinator. They can tell you exactly what format visits take, what the current schedule is, and what you need to bring or do to be approved as a visitor.