Without a photo we can only make an educated guess; I suspect a "lichen". Lichen comes in different colors and growth habit; it grows on the surface and is not considered a disease, therefore there is no reason to remove it.
There is a slight chance it is a fungus disease, but I can't think of one that appears red on privet branches. There leaf spot diseases but fungus on the stems would be wood rot. If it's growing on exposed heartwood from a wound or pruning cut then it could be a wood decay fungus.
I'm not sure what you mean by "amoeba-like", unless you mean that it appears as formless and spreading on the surface, which could apply to lichen or fungi. Amoeba are microscopic organisms so if you can see it, it's not ameoba.
A picture is worth a thousand words you know. We don't always know the identification, but a photo sure helps. Don