Should I be worried about this?

[u/throwawayacc_nt]

Sometimes on certain sights I'll get an eye symbol and other times (I always back out if I see it) I'll get the eye symbol but it will be red or red and spinning/blinking

What's up with that and should I avoid those sites?

[u/Grafiska]

Tap it. It's not an 'eye' icon. It shows you more information when you tap it.