It can depend on which Jabber server you are using and what has been enabled on that particular Server.
That's one possible.
Another is that the Applications at the other end is not set up to accept File Transfers, such as someone logged in to AIM Express in a web browser will have limited abilities.
Finally, AIM file transfers are on a different port and or internet protocol and this port (and or the Protocol) may not be allowed though your Mac settings in.
Lets start with the last.
AIM Logins in version up to and including version 5 are on port 5190 by default using the TCP Protocol.
We (The regular posters) have been suggesting using port 443 for this for a long time. Now iChat 6 and Messages in Mountain Lion actually use port 443.
File Transfers on on Port 5190 but use the UDP Protocol.
It always uses this and this cannot be changed.
In the Mac Firewall in some versions of the OS UDP traffic can be blocked.
Check the Advanced Button settings.
Of course the port has to be opened in the modem (if it routes) or your router.
In some cases such as the Port Forwarding method you can have models that specify Protocol as well as ports. Opneing it for 5190 and TCP can block it being opened for 5190 and UDP.
If you are using UPnP (Or Apple's Port Mapping Protocol) you should not have problems.
With the second option above you can pull down the Video Menu and open Connection Doctor.
This will tell you your Capabilities.
In a Buddy List highlight a Buddy.
Use the CMD and i keys together (Get Info)
The Capabilities tab in here will tell you whether the Buddy has the abilities to File Transfer.
The first option I suggested can only really be checked using a Third Party Jabber app such as Psi
Basically after you set up an Account in Psi you can access an option called Service Discovery that iChat (and Messages in Mountain Lion) does not do.
This will tell you if the server is set up for File Transfers or Group chats plus things called Transports.
With some servers you will need to (sub) register for the Services you also want to be active.
Using this method to view Google's GoogleTalk server tells you nothing about the Services they have active.
As you may have worked out you may need to check out the server's ability if you are using a Jabber servers even if the symptom looks more like reason two.
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Mountain Lion 10.8.2)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad
"Limit the Logs to the Bits above Binary Images." No, Seriously