This isn't a simple answer, but it's my best guess.
There's not much in our world today that you can do that doesn't involve liability. Simply driving down the road makes you liable for things.
To answer your question, though, you can take a look at what establishes tort. Three questions are usually asked in a liability case:
1) there must be a duty to care to another person
2) there must be a breach of that duty
3) the claimant must have suffered damages caused by the breach of duty.
As a professional engineer, your duty to care is elevated when you establish that your a professional engineer, because you are using your license to establish yourself a competent person. In your case, you would have a duty to care for the drawings your checking.
(For example, your client hires a 3rd grader instead of you, it would be hard to establish that the 3rd grader has a duty to care in code reviews since it's something that the 3rd grader is not knowlegable in. On the other hand, you, as a professional engineer, are supposed to know something about it.)
You can breach that duty by making mistakes, which we all know happen often as an engineer. Engineers live in a black and white world while restuarant workers and actors live in a very colorful world.
Your client could suffer damage if they lose a bid (for example) because their drawing proposal doesn't meet code and you attested that it does.
So I think the answer is yes-you do assume liability. There are ways you can minimize the liability but I'll hold off on suggesting since that wasn't your question.