The TNG Technical Manual indicates that the command authority to signal alert conditions (or give any shipwide orders) ultimately rests with the vessel's Commanding Officer, typically a captain, and, by extension, their First and Second Officers. Additionally, Red and Yellow alerts can also be triggered by event conditions (such as the failure of a critical system or someone pointing lasers at you) although these can be overriden by the commander if there's a good reason to do so.
Within the bridge itself, it's the Operations (Ops) Manager who is responsible for physically setting the ship to condition red or yellow but the computer can also be ordered to do this by one of the ship's command staff and can also do this automatically. The next order is usually to inform the Captain (or most senior officer) that their presence is required on the bridge.
Operational authority for the starship rests with the Commanding
Officer (usually the captain or duty officer). The Commanding Officer
is responsible for execution of Starfleet orders and policy, as well
as for interpretation and compliance with Federation law and
diplomatic directives. As such, the Commanding Officer is directly
answerable to Starfleet Command for the performance of the ship.
The Main Bridge is directly responsible for the supervision of all
primary mission functions. Through the Operations Manager, the bridge
also monitors all secondary mission functions to provide an optimal
operating state. The multimission operational profile of the
Enterprise requires extensive coordination between different
By comparison, the command ....
... can be given by any member of the crew and can be done verbally or via a comm panel. The ship may also declare an intrusion alert on its own without a crewman being present.
This alert signals to the bridge Chief of Security that an intruder has been spotted which will, presumably, result in the arrival of a security team as well as informing the bridge that an intruder is suspected.