I would like that my Garmin 530 pop-up a segment when I am on it. I don't have Strava premium, so I was thinking of the same functionality but from Garmin. I have defined some segments and marked them as favorites
While I'm having no issues with Strava Live Segments at all, Garmin segments should work the same.
Once they are sent to your device, your PRs will get updated on the segments if you don't forget to synchronise the EDGE unit with Garmin Connect App, via PC or WIFI. I don't think the unit actually updates your PRs on the segment files on the device, and that is done via connect and pushed back to your device on the next sync.
I was riding a segment multiple times yesterday, and my PR time stayed the same according to the device, and consequently all three efforts I was always beating my PR according to the device. Only after getting home and uploading my ride, did the device get segments updated with my new PR.
The site that manages the segment is the arbitrator on whether you set a new PR or and other record such as a KoM.
This requires that your activity be uploaded so that the site can process it and update the leaderboard data and then send the updated leaderboard back.