You can actually dimly perceive near-IR LEDs -- they'll glow slightly red in darkness.

That depends on how "near" they are.

The sensitivity to red light decreases quickly at wavelengths greater than 650 nm, but light can still be perceived if it is strong enough, up to around 780 nm.

Many so-called near-IR LEDs may actually be somewhere around 750 nm, so they are still visible on a dark background, even if they are perceived as extremely dim.

On the other hand, there are many near infrared LEDs around 900 nm and those are really invisible. Near-infrared LEDs around 1300 nm or around 1550 nm are also completely invisible.

An invisible near-infrared laser beam could become visible due to double-photon absorption, but if a beam of such intensity as to cause double-photon absorption hits your retina, there are more serious things to worry about.

I remember reading some people can perceive some near IR, but mostly that near-IR LEDs actually leak some red themselves due to imperfections in manufacturing or something?