We see Jessica Fletcher constantly ask the local Sheriff's and homicide detectives working with her (or the Sheriff's and detectives themselves) to put out APB's (All Points Bulletins) or BOLOs (Be On the Look Out) to bring suspects and witnesses in. When found the police just walk up to people and place them in the squad car, taking them to the station for Mrs. Fletcher and the Sheriff's/detective's to interview even though it's pretty clear they didn't want to come. If there is no probable cause for an arrest or an active arrest warrant, the police can't make you go anywhere against your will. Once in a while, a wealthy or educated person will assert this and ask a lawyer to be present but this is rare.
Jessica Fletcher often undertakes actions which would result in evidence and investigation being compromised. Since she is only a crime novel writer and has never been a police officer, these actions could result in cases being dropped or overturned upon appeal. There is no way that the local District Attorney would allow her to perform the actions that she does.