You've got it right in that the speed going up, at least at the start, is
determined by the muzzle velocity of the bullet.
The speed down is determined by the terminal velocity of the bullet, which
this "Straight Dope" article addresses.
"A bullet fired straight up will slow down, stop, then fall to earth again,
accelerating until it reaches a point where its weight equals the resistance
of the air. That's its terminal velocity."
The "Go Ask Grandpa" website goes into more detail on muzzle velocities and
presents some of the numbers involved.
"The bullet leaves the gun at 3000 feet per second. There are two things
now that slow the bullet down as it goes up. One is gravity - which slows
the bullet 32 feet per second every second. ... air friction is slowing the
[The] terminal velocity is different depending mostly on two things. 1.
The density of the thing that is falling. 2. The shape of the thing that
is falling. ... The terminal velocity of a feather may be only 2 feet per
second because of its shape and density, but the terminal velocity of a
bullet could be 300 or 400 feet per second."
So, unless your chum is talking about a bullet in a vacuum, then the speed
of a bullet is much slower, but still dangerous, on the way down than on
the way up.
If you need any clarification, please feel free to ask.
Google search on: bullet up down "same speed"
Looking Forward, denco-ga - Google Answers Researcher