They would not have given the Miranda Rights speech, as that wouldn't become policy until the mid-1900s, if not later. It would depend on where in this century you're talking about, as I'm not sure if police work in, say, 1812 was the same as police work in 1892.
This might sound a bit daft, but watch Sean Connery in The Untouchables; he's like the quintessential copper in them days.
From the thread title, I thought this was about some new hotline. 'Call to talk to a sexy constable now!'
Officer Olivia is standing at attention to take your call. But watch out, she might have to frisk you if you're naughty ... *giggles*
Well I went ahead with something I saw on yahoo that sounded right. I've already posted it for review.