Corker Binning, reportedly a law firm with good reputation in the field of handling extradition representation, describes a red notice as
... an alert issued by Interpol at the request of one of its member countries, indicating that the country seeks the individual’s provisional arrest with a view to extradition. Red notices are recorded on Interpol’s databases for instant circulation to police forces and border agencies around the world.
This description is presumably correct, if perhaps incomplete. A key term in this description is extradition - the removal of a person, in custody, from one state to another state.
This can cause problems for the application by local law enforcement.
A British man was in 2009 the subject of a red notice. Having left Dubai for personal reasons, he was reported for fraud based on the defaulting of a car loan, but
... despite featuring in Interpol's list of the 45 most wanted Britons, he was told by officers there was no outstanding warrant against him in the UK and therefore could not be arrested.
This suggests that at least for citizens of the United Kingdom, it is not possible to arrest the red notice subject for that reason alone.
More generally, Interpol cannot require action (arrest or otherwise) on the basis of a red notice. They are for information purposes only. This is pointed out by Interpol themselves, when they say that
INTERPOL cannot compel any member country to arrest an individual who is the subject of a Red Notice. Each member country decides for itself what legal value to give a Red Notice within their borders.
Given the above story and several others online of police declining to arrest red notice subjects without significant political pressure, it does not appear an individual subject to a red notice can be arrested for that reason alone.