The physical effects of the Buzz are constant: the Immortal (usually) feels a prickling at the base of his or her neck, followed by what appears to be a surge of adrenaline. Physical senses appear to be enhanced, to the point where one Immortal can turn and zero in on the other without even looking. For a newly created Immortal, the experience can be unsettling, even traumatic as seen by Claudia Jardine's reaction to both Walter Graham and Methos at various points.
The effects of the Buzz may vary between immortals but are usually focused on the cranium and usually enough to be immediately noticeable. There seems to be a different effect when an Immortal meets a latent Immortal (Immortals before their first death) that makes them recognizable to Immortals as hinted at by Conner MacLeod after meeting Richie Ryan. Latent Immortals don't feel the Buzz.