So I think I have a complete grasp on the question, and IBE. I'm going to attempt an answer. There is an end to my madness.
IBE is based on bilinear pairings in cryptography. Page 23 of that paper is where they start talking about pairings, but I think the whole paper is relevant. A pairing scheme often used by IBE is Weil Pairing (also described in that document). These are based in elliptic curves math magic.
Elliptic curves are a lot like Diffie Hellman in that they're both based in the discrete log problem. Here's a question and answer as to why DH can not be used for digital signatures. There are some digital signature schemes that use Elliptic curves as a base, but additions are needed (such as hashing algorithms). ECC Based Digital Signature Schemes.
My line of thinking is that the statement "Alice encrypts with her private key." is not as simple as it seems. Like Elliptic curves and DH, a pairing can be generated for a given input. But you first need to have the input in order to get the generated private key. Which implies that encrypting with the private key would not result in a successful decrypt by the public key. In the same way that you have separate decrypt and encrypt functions in some modes of symmetric encryption.
I'm not saying that you can't provide a means for this. There are standards out there for ECC based IBE Signatures (RFC 6507), but they have to be built into IBE scheme for such verification to work.
I learned something in all this, my previous comment about public key cryptography concepts generally the same was completely wrong. I hope this answers your question. I tried to keep Wikipedia articles to a minimum :)