Yes - you can visually check that the exhaust bypass valve is operating.
The valve actuator is visible to the left of the main silencer box - as viewed from the rear of the car - through the mesh immediately above the rear diffuser. You should see what looks like an end-on piston that operates the exhaust valve through a crank linkage. The valve should default to "open" against spring pressure; when activated (i.e. "Valve closed"), the piston is pulled-in by applied vacuum.
What you can't visually check is operation of the vacuum solenoid valve that (for the Exige S) is located on the rear left of the engine subframe - mounted vertically below the expansion bottle in the engine bay. This valve has been known to fail due to water contamination. This valve is not present on the Exige 350 Sport.
Don't be tempted to diagnose solenoid operation by disconnecting it from the wiring harness. If you do disconnect when the engine is running or prior to engine start, the ECU will sense an electrical/exhaust fault and illuminate the Engine Warning lamp - which will likely require reset by a Lotus dealer. Only diagnose potential faults in the vacuum system by selectively disconnecting the vacuum hoses from the solenoid. The upstream side connects to the vacuum reservoir via a t-peice in the vacuum line. The downstream vacuum hose connects to the exhaust valve actuator - and vents to atmosphere via a third (enclosed) port under a cap on the solenoid.