I used to teach a class of 12 kids between 9 months and 2 years (with two other wonderful teachers), so I know what you're going through and then some. It is always a challenge at this age. I think you got some bad advice from well-meaning moms. Most moms didn't say what they meant by discipline, but most of their suggestions are not appropriate for 12-month olds. If you hit them, they will associate the hitting with you, not their behavior. Time out is useless for kids who can barely walk. Again, they will not associate the punishment with the behavior, and their attention span is way too short for them to "sit and think about what they've done".
At this age, even twins don't really play "together", but they play side by side. Because it doesn't seem to be bothering the less aggressive twin, you could choose to ignore it. Since it's bothering you, I would just take the toy away and give it back to the first. Then spend a minute playing with the less agressive twin and the toy that was taken away. Giving one attention and affection when they are displaying an appropriate behavior and ignoring inappropriate behavior works much better at this age.
You could try getting two of everything. I have heard some moms of twins say that is essential. On the other hand, it's probably part of the fun to take the toy away, so this will not stop the behavior entirely.
With 12 toddlers, this was a constant occurrence. For the most part, we learned to ignore it. With the littler kids, we tried to avoid toy sharing to avoid germ sharing, but that's impossible between twins. I would try to keep your kids happiness in mind. If the kids are happy playing, then play happily with them. "Sharing" and "fairness" will come up soon enough. These are important values to teach all siblings, but it will be much more challening with twins.
Good luck, and try to enjoy this exciting time with your boys!