can I find out if someone claimed my child on their income taxes and also can I find out if they got a tax return.

No, not unless you file yourself. The IRS also will not tell you how much refund was paid, if any.
They will just decide which one of you had the right to file, and penalize the other one.

The way you find out is by claiming the child yourself. If someone has also claimed the child the IRS will contact you and try to resolve who is entitled to claim the child. They will not, however, tell you who the other filer was or what their refund was.