把 structure should be accepted, it is more correct as you are asking for a transfer. 你可以把一张纸给我吗？
You're right, it's 把 structure not used at all?
Actually 把字句【把 structure】and 被字句【被structure】can be changed into each other. For example：
【Can this paper be given to me？】
BUT the latter is quite weird for Chinese natives......
Why is there no 你？
This is a common phenomenon in Chinese sentence. Just remember this structure. When you use oral Chinese, you can use this structure. 你 is added to make this sentence more formal.
I had 我可以借一张纸吗 …
Does ”可以你给我一张纸吗“ work? Or would you need to have 你 before 可以?