Do we hear sounds differently on the highest mountains?

There more sides to this scenario that you're considering. Firstly, if we are assuming that the temperature is the same at sea level and on the high mountains, then the speed of sound doesn't actually change, as a constant temperature will take care of the air pressure-density ratio. $$c = \sqrt{\kappa \frac{p}{\rho}} $$ Where $p$: static air pressure, $\rho$: air density and $\kappa$ the adiabatic index $c_p/c_v$.

Again the statement being: The static air pressure and the density of air are proportional at the same temperature, meaning the ratio $p/\rho$ is always constant, on a high mountain or even on sea level altitude.

So in the scenario you're describing, if $T$ is taken as constant, then the speed of sound doesn't change, but its intensity does, as the density of the air is much lower on top of mountains, rough approximation for the intensity would be $$I \propto p v \propto \omega² c \rho$$ Where $v$ is the speed of air molecules, $\omega$ sound frequency. So one thing is for sure, you will need to shout much louder on a mountain, for people further ahead to be able to hear you.

Furthermore, if $T$ is changing, then ($p/\rho=constant$) doesn't hold anymore, so $c_{air}$ changes. There are rough approximates relating the speed of sound to $T$. In a crude manner: $$c_{air} = 331.3 \frac{m}{s} \sqrt{1+\frac{\theta}{273.15}}$$ Where $\theta$ is the air temperature in °C. Such estimate gives e.g. $60 \frac{cm}{s}$ change of speed of sound for $1 °C$ change of temperature. Further scenarios:

At $-20°C$: $c_{air} \approx 319 \frac{m}{s}$

At $0°C$:$c_{air} \approx 331 \frac{m}{s}$

At $20°C$: $c_{air} \approx 343 \frac{m}{s}$

At $100°C$: $c_{air} \approx 387 \frac{m}{s}$

Next logical step would be to consider the change in wavelength of the sound, when $c_{air}$ changes. For this you have the general formula $c = \lambda f $

So $\lambda$ changes with $T$ as the speed of sound changes and in case of a flute e.g. the length of the vibrating air column doesn't change, so when $c_{air}$ changes due to $T$ fluctuations, then the sound frequency $f$ changes (or pitch of tone as you call it). But since we don't use flutes to speak, this doesn't concern us, so in order to conclude, in a mountain, the intensity I, speed of sound $c$ and the sound wavelength $\lambda$ change (the last two only hold for $T$ varying) but not the pitch.

I'll use this answer to provide some information that's mostly orthogonal to what Phonon said.

As Phonon pointed out, the speed of sound depends on temperature, not pressure. It's cold on the top of high mountains, so the speed of sound would tend to be lower. Some mechanisms for sound production have a frequency that depends on the speed of sound, and others don't. In the former category, we have wind instruments, which act like resonant air columns, and the human vocal tract, which is complicated but can be understood to zeroth order as a Helmholtz resonator. So one's voice is lower when the air cold, and I think this fact is well known to, e.g., professional singers. In an orchestra, we have some instruments that act like air columns (brass and winds), and others whose pitch is roughly independent of air temperature (strings). As an orchestra plays, the wind instruments warm up and their pitch rises.

Do humans notice these differences in sound? Does the pitch of tones change noticeable?

The highest I've ever been is 5900 m. It was certainly cold, so most likely the pitch of our voices was a little lower than normal, but it wasn't noticeable.

The other effect that one might expect to notice would be that sounds would appear more faint because of the poorer coupling of the radiator to the air and of the air to the receiver (e.g., eardrum). This has nothing to do with temperature or frequency. An easy example to imagine is a speaker. The speaker has a surface (the "speaker cone") made out of something like cardboard or rubber. As the cone vibrates, it excites vibrations in the air. If there's less air, then this excitation is less efficient, because there is less for the speaker to push against. The same thing happens at the eardrum or at the sound-sensitive surface of a microphone. For these reasons, one might expect that human voices would appear softer than normal at high altitude.

The reality is that this has not been a noticeable effect at any altitude I've ever been to. Basically it tends to be windy at high altitude, especially on passes and summits, so if it's windy, it's simply hard to hear for that reason. If the air is calm, then these are very quiet places because they're out in the wilderness, so you can easily hear over long distances.

I'm sure it's true that on the top of a tall mountain the intensity of the sound waves from people's voices is down by several db compared to normal. However, I think there are two facts about the ear-brain system that make this is not noticeable. (1) The physiological sensation of loudness is extraordinarily compressed, in the sense that your ear-brain system takes many orders of magnitude worth of sound intensities and makes them into a perceptual range that doesn't seem so broad subjectively. (2) Psychological sensations are easy to judge when there's an immediate comparison, i.e., in a relative context, but much harder to judge in absolute terms. Think of the eye exam where they flip lenses in and out and ask which is clearer.

One tends to notice these effects very clearly in the classic classroom demonstration where we play a sound through a loudspeaker from inside a bell jar and pump out the air. The effect is very hard to notice until the vacuum inside the bell jar gets quite good (maybe 90-95% vacuum). When I do this for a room full of students, they generally don't notice any difference at all, even at 95% vacuum, until I quickly let the air his back in, and then they can hear the relative loudness more easily because it's a rapid comparison.