Then it occurred to me... if MAMP was working, maybe there was something in its configuration that I needed. So I set about looking through MAMP's httpd.conf and found this:
<IfModule !mpm_winnt.c>
<IfModule !mpm_netware.c>
#
# If you wish httpd to run as a different user or group, you must run
# httpd as root initially and it will switch.
#
# User/Group: The name (or #number) of the user/group to run httpd as.
# . On SCO (ODT 3) use "User nouser" and "Group nogroup".
# . On HPUX you may not be able to use shared memory as nobody, and the
# suggested workaround is to create a user www and use that user.
# NOTE that some kernels refuse to setgid(Group) or semctl(IPC_SET)
# when the value of (unsigned)Group is above 60000;
# don't use Group #-1 on these systems!
#
User myusername
Group #-1
</IfModule>
</IfModule>
Copy+paste that into /etc/apache2/httpd.conf and voilá! Permission problems solved.
