Typically N-MOSFET is used for low side switching and P-Type for high side switching. That's why transmission gates use both N and P-type MOSFET.
Actually, NMOS is also often used for high-side switching in cases of higher supply voltage which would require extra gate-drive circuitry for a PMOS anyways since pulling the PMOS gate all the way to ground would exceed its maximum tolerable Vgs. If you need extra gate-drive circuitry anyways, you might as well just use a cheaper, more efficient, more available NMOS on the high-side which also requires gate drive circuitry.
But I can't correctly understand how a P-Type for the low side and N-Type for high side switching creates degraded output. It would be nice if someone could explain this to me briefly.
What you are looking at there is called a "source follower". More specifically, a PMOS source follower and an NMOS source follower stacked on top of each other.
It's not really acting as a switch. It's a source-follower being pressed into service as a switch. So it's not very efficient and not for high power. But that configuration is chosen when only low power is involved and you do not want extra circuitry to deal with shoot-through since the source-follower inherently cannot shoot-through when both high and low-side MOSFETs gates are tied together and driven from a common signal. You also find it amplifier outputs which should give you a better impression of what it really is since it can drive signals at partial levels between on and off.
You can look up what a source follower is but here is a summary of what is relevant:
Remember, Vgs controls how much the MOSFET conducts. Vgs, the voltage between the gate AND source terminal. Not Vg (gate voltage referenced to ground). In a source follower, the source terminal is not connected to ground, but the gate is being driven by a voltage referenced to ground. As the MOSFET conducts more and more the voltage drop across the load causes Vs (source terminal voltage referenced to ground) to change. Since Vg is being driven relative to ground, this changing Vs moves towards Vg which reduces Vgs thereby preventing the MOSFET from completely conducting. An equilibrium is reached between full off and full on.