Do two objects (two large hunks of lead suspended from two strong strings) exert a gravitational force on each other? If so, why cant we see them attract each other?

Because the gravitational constant G is 6.7 x 10^-7

And it makes the gravitation force relatively small in terms of magnitude.

At least until it doesn't come to the objects with the huge masses (like planets or moons with their 10^20 masses).