You'll need to store the time zone along with the local time, otherwise you won't be able to handle the situation where you have multiple users in different time zones. About the only reason this is preferable to storing UTC along with time zone, is because there are sometimes political decisions made to change daylight savings time (i.e. the mere fact of DST/TZs etc. isn't enough).
Yes, I store the timezone (associated with the user) along with the local time. All date operations in the application are done related to their timezone.
Storing it this way is preferable because it makes working with times across DST boundaries a non-existent problem. DST is a pain in the ass and you want to use whatever language / database facilities to exist to make this as smooth as possible -- storing local time (and setting the timezone appropriately) makes it not your problem. Also users frequently set their own timezone incorrectly, and when they fix their timezone, they don't want all their appointments to be at the wrong time.
