If I'm understanding segwit correctly, miners have to verify signatures in order to include a transaction into a block that they are mining. However, miners are not required to verify signatures in order to decide that a block a different miner mined is valid. Is this correct?
Miners can do what they want to construct blocks, including creating blocks with invalid signatures. They can even try construct blocks by just creating random data, and hoping it's valid.
However, blocks without valid signatures (or other validity problems) will not be accepted by the network. All full nodes verify all signatures in all blocks, so if a miner constructs a block with an invalid signature, they've just wasted a enormous amount of effort to produce something everyone will just ignore, and forgone the ability to be paid for that block. The same is true for building on top of an invalid block; thus miners are incentivized to validate the blocks they build on top of too - but they don't have if they like losing money.
None of this has anything to do with segwit.