Why is Lamport-Diffie secure? Why is Lamport-Diffie secure? I note that there is a demonstration based on onewayness (in the book postquantum cryptography). But a one way function is not sufficient to ensure that it can not infer some bits or any feature of the key signature.
Lets assume an adversary knows one LD signature for message $M$ as well as public key $pk$ and can generate a forgery for an arbitrary different message $M'$. Clearly, there exists at least one bit position where the messages differ, i.e. $M_i \neq M'_i$, as $M \neq M'$. So, for simplicity assume there is only one bit difference. Then the adversary can take all signature values for the signature on $M'$ from the signature on $M$ besides the value for this single bit position. For the signature value of this bit it must hold that $f(sig_i) = pk_{i,M'_i}$ as otherwise verification would fail.  

So if the adversary is able to generate a valid forgery it obviously is able to find a value $x$ that satisfies $f(x) = pk_{i,M'_i}$ for given $pk_{i,M'_i}$. This is per definition finding a preimage of $f$. 

To get back to your question: The adversary might learn some information about $sk_{i,M'_i}$ from $pk_{i,M'_i}$. But that is not enough to produce a forgery as a forgery requires knowledge of a complete value $x$. Even more: it is not required that $x = sk_{i,M'_i}$ for the forgery to be valid.