Yes but only over 'parallel' sections of route, basically between the Bus Station and Park Barn (via the University or The Chase) and between the Bus Station and Curry's/PC World at Woodbridge Meadows (via Woodbridge Road). Some Arriva drivers are a bit puzzled when presented with a Safeguard weekly in my experience, however, although I have actually never been refused travel with one.
Note that Day Tickets are NOT inter-available and that neither company's tickets are valid on Stagecoach, even though their Route 20 runs parallel with Arriva and Safeguard between the Bus Station and Woodbridge Hill/Southway.