When your developer runs git-push into the repository, git-receive-pack is run (either locally or over ssh) as that developer, so is hooks/update script. Quoting from the relevant section of the documentation:

Before each ref is updated, if $GIT_DIR/hooks/update file exists
and executable, it is called with three parameters:
$GIT_DIR/hooks/update refname sha1-old sha1-new
The refname parameter is relative to $GIT_DIR; e.g. for the
master head this is "refs/heads/master".  Two sha1 are the
object names for the refname before and after the update.  Note
that the hook is called before the refname is updated, so either
sha1-old is 0{40} (meaning there is no such ref yet), or it
should match what is recorded in refname.

So if your policy is (1) always require fast-forward push (i.e. never allow "git-push repo +branch:branch"), (2) you have a list of users allowed to update each branch, and (3) you do not let tags to be overwritten, then you can use something like this as your hooks/update script.

[jc: editorial note. This is a much improved version by Carl since I posted the original outline]

#!/bin/bash

umask 002

# If you are having trouble with this access control hook script
# you can try setting this to true.  It will tell you exactly
# why a user is being allowed/denied access.

verbose=false

# Default shell globbing messes things up downstream
GLOBIGNORE=*

function grant {
  $verbose && echo >&2 "-Grant-         $1"
  echo grant
  exit 0
}

function deny {
  $verbose && echo >&2 "-Deny-          $1"
  echo deny
  exit 1
}

function info {
  $verbose && echo >&2 "-Info-          $1"
}

# Implement generic branch and tag policies.
# - Tags should not be updated once created.
# - Branches should only be fast-forwarded unless their pattern starts with '+'
case "$1" in
  refs/tags/*)
    git rev-parse --verify -q "$1" &&
    deny >/dev/null "You can't overwrite an existing tag"
    ;;
  refs/heads/*)
    # No rebasing or rewinding
    if expr "$2" : '0*$' >/dev/null; then
      info "The branch '$1' is new..."
    else
      # updating -- make sure it is a fast-forward
      mb=$(git merge-base "$2" "$3")
      case "$mb,$2" in
        "$2,$mb") info "Update is fast-forward" ;;
        *)        noff=y; info "This is not a fast-forward update.";;
      esac
    fi
    ;;
  *)
    deny >/dev/null \
    "Branch is not under refs/heads or refs/tags.  What are you trying to do?"
    ;;
esac

# Implement per-branch controls based on username
allowed_users_file=$GIT_DIR/info/allowed-users
username=$(id -u -n)
info "The user is: '$username'"

if test -f "$allowed_users_file"
then
  rc=$(cat $allowed_users_file | grep -v '^#' | grep -v '^$' |
    while read heads user_patterns
    do
      # does this rule apply to us?
      head_pattern=${heads#+}
      matchlen=$(expr "$1" : "${head_pattern#+}")
      test "$matchlen" = ${#1} || continue

      # if non-ff, $heads must be with the '+' prefix
      test -n "$noff" &&
      test "$head_pattern" = "$heads" && continue

      info "Found matching head pattern: '$head_pattern'"
      for user_pattern in $user_patterns; do
        info "Checking user: '$username' against pattern: '$user_pattern'"
        matchlen=$(expr "$username" : "$user_pattern")
        if test "$matchlen" = "${#username}"
        then
          grant "Allowing user: '$username' with pattern: '$user_pattern'"
        fi
      done
      deny "The user is not in the access list for this branch"
    done
  )
  case "$rc" in
    grant) grant >/dev/null "Granting access based on $allowed_users_file" ;;
    deny)  deny  >/dev/null "Denying  access based on $allowed_users_file" ;;
    *) ;;
  esac
fi

allowed_groups_file=$GIT_DIR/info/allowed-groups
groups=$(id -G -n)
info "The user belongs to the following groups:"
info "'$groups'"

if test -f "$allowed_groups_file"
then
  rc=$(cat $allowed_groups_file | grep -v '^#' | grep -v '^$' |
    while read heads group_patterns
    do
      # does this rule apply to us?
      head_pattern=${heads#+}
      matchlen=$(expr "$1" : "${head_pattern#+}")
      test "$matchlen" = ${#1} || continue

      # if non-ff, $heads must be with the '+' prefix
      test -n "$noff" &&
      test "$head_pattern" = "$heads" && continue

      info "Found matching head pattern: '$head_pattern'"
      for group_pattern in $group_patterns; do
        for groupname in $groups; do
          info "Checking group: '$groupname' against pattern: '$group_pattern'"
          matchlen=$(expr "$groupname" : "$group_pattern")
          if test "$matchlen" = "${#groupname}"
          then
            grant "Allowing group: '$groupname' with pattern: '$group_pattern'"
          fi
        done
      done
      deny "None of the user's groups are in the access list for this branch"
    done
  )
  case "$rc" in
    grant) grant >/dev/null "Granting access based on $allowed_groups_file" ;;
    deny)  deny  >/dev/null "Denying  access based on $allowed_groups_file" ;;
    *) ;;
  esac
fi

deny >/dev/null "There are no more rules to check.  Denying access"

This uses two files, $GIT_DIR/info/allowed-users and allowed-groups, to describe which heads can be pushed into by whom. The format of each file would look like this:

refs/heads/master   junio
+refs/heads/pu      junio
refs/heads/cogito$  pasky
refs/heads/bw/.*    linus
refs/heads/tmp/.*   .*
refs/tags/v[0-9].*  junio

With this, Linus can push or create "bw/penguin" or "bw/zebra" or "bw/panda" branches, Pasky can do only "cogito", and JC can do master and pu branches and make versioned tags. And anybody can do tmp/blah branches. The + sign at the pu record means that JC can make non-fast-forward pushes on it.