Do I need common trip breakers for a house wired with shared neutrals?
The National Electrical Code requires simultaneous disconnection of multiwire branch circuit ungrounded conductors.
It does NOT require common trip breakers.
Here are the pertinent articles:

210.4(B) Disconnecting Means. Each multiwire branch circuit shall be provided with a means that will simultaneously disconnect all ungrounded conductors at the point where the branch circuit originates.
Informational Note: See 240.15(B) for information on the use of single-pole circuit breakers as the disconnecting means.
240.15(B) Circuit Breaker as Overcurrent Device. Circuit breakers shall open all ungrounded conductors of the circuit both manually and automatically unless otherwise permitted in 240.15(B)(1), (B)(2), (B)(3), and (B)(4).
(1) Multiwire Branch Circuit. Individual single-pole circuit breakers, with identified handle ties, shall be permitted as the protection for each ungrounded conductor of multi-wire branch circuits that serve only single-phase line-to-neutral loads.

Notice the last Article 240.15(B)(1) allows you to use identified handle ties on single pole breakers for multiwire branch circuits.
Besides that, any house built under an older Code (like 1983) is normally grandfathered until you make major changes, like replacing the panel, and have to then upgrade to the new code requirements.
As far as determining which circuits are multiwire. You can start with the white and black on the 50 amp breakers those should be paired. Then any red wire would normally be in a multiwire cable with a black. Determining which red goes with which black will be your project.
Good luck and stay safe!