From GMAT Prep

(1) says the product of any two numbers on the list is 0

Lets select a list say {0,0,0,2} here the product of any two numbers would be 0 but all numbers do not equal 0. Now lets select another list say {0,0,0,0} then the product of any two numbers would be 0 and all numbers equal 0.INSUFFICIENT.

(2) says the sum of any two numbers on the list would equal 0.This can be true only if all the numbers equal 0.For example lets select a list say {0,-2,2,0} here -2+2=0 but 2+0=2 .Try plugging in more numbers and you will see that the sum will not equal 0 unless all numbers are 0.Hence SUFFICIENT.

So the answer is B

Correct me if I am wrong

When the problem mentions "the sum of any 2 nos." in statement 2, does that mean all numbers should ultimately give the sum as 0 when added to any other no.?
Example - (2,-2,2,-2) will give 0 when 2 and -2 are added but not when 2 and 2 are added.

Exactly!! It means that when you add any two numbers in the list the result should ultimately be 0

moneyman wrote:(1) says the product of any two numbers on the list is 0

Lets select a list say {0,0,0,2} here the product of any two numbers would be 0 but all numbers do not equal 0. Now lets select another list say {0,0,0,0} then the product of any two numbers would be 0 and all numbers equal 0.INSUFFICIENT.

(2) says the sum of any two numbers on the list would equal 0.This can be true only if all the numbers equal 0.For example lets select a list say {0,-2,2,0} here -2+2=0 but 2+0=2 .Try plugging in more numbers and you will see that the sum will not equal 0 unless all numbers are 0.Hence SUFFICIENT.

So the answer is B

Correct me if I am wrong

you're right on the money on this one.