Pretty clear each party is responsible for 50%  either financially or physically.  Unfortunately if you read the annotation and go to the Fiene v. Robertson, 184 Neb. 668, 171 N.W.2d 179 (1969) case you'll see that even if the other guy doesn't do anything with the fence the one with the cattle is 100% liable for any trespass and destruction of property.

"It appears from the record that defendant may have had a just grievance occasioned by plaintiff's failure to maintain a just proportion of the division fence as contemplated by section 34-102, R.R.S. 1943. Nevertheless, this did not justify or excuse the trespass by defendant's cattle. On the contrary, defendant's remedy was to compel plaintiff to maintain the fence in the manner provided by Chapter 34, Article 1, R.R.S.1943."

So yes they both have to contribute equally but if you build half a fence and your cattle go do $100K in damage on the other side you are liable even though the other guy didn't do anything, so in other words it completely neuters the law.  Best case is probably build 100% and sue for reimbursement of 50%.  At least that way you get exactly the fence you want.