I agree, when I first got my Fitbit all worked well. My phone would ding, the message would appear on my fitbit. Now notifications don't work on my phone and my fitbit doesn't display the texter or text information. My notifications are on, my alert noise is set, I have not hidden alerts for any of my contacts, my mute button is not engaged. As I play with these settings I will always get an alert for the first text, but none of the subsequent messages. Today I tried to disable blue tooth and I got the notification. Oddly enough after disabling bluetooth I also noticed Do not disturb was on, so I disabled that and re-enabled bluetooth. I got the notification, but the text alert was not sent to my fitbit like it used to. I will need to wait to see if I get alerts for any more texts or if it's still only working for the first text after a change.
FYI - My fitbit also isn't syncing properly, my dashboard on the web is correct, but the Fitbit app doesn't display my sleep information correctly. All days show the current day's information. Could be the Fitbit app messing things up. I"ve posted to Fitbit forums about the sync issue and nobody was able to help me (so far).