If this were with fraternal (non-identical) there would be few similarities (since fraternal twins don't necessarily share the same chromosome profile). However, this would be interesting were the twins identical. Both fathers would be totally identical genetically to each other, as would the mothers. However, this does not mean that each father would necessarily pass on the same genetic information as their twin.
For example, male twin 1 could pass on his X chromosome, while male twin 2 could pass on his Y chromosome, meaning the children would be of opposite genders. This type of pattern would follow for every trait that each set of twins was heterozygous for, meaning that we would probably see a great deal of difference between the two children.
So, to answer your question- the children's DNA would likely be more similar than any two other random children, but by no means identical. And the children might show some resemblance, but it would be highly, highly unlikely that they would be identical.
If they are identical twins (formed from the same egg so that their DNA is identical) their babies are more likely to resemble each other. Their DNA would be similar. If both sets are fraternal twins (2 separate eggs fertilized) the babies are no more likely to resemble each other than in non-twin births.