"Can multiprocessing be carried out using 8031(or 8051)"

Of course it can!

"i.e sharing of a memory chip by 2 8031/8051 ucontrollers"

There are many ways to implement multiprocessing without necessarily using shared memory.

"Do I have to use a DMA controller for the same?"

No.
eg, you could use dual-port RAM:
http://www.8052.com/forum/rea...&top=54235