By the time you reach the river in an Omaha 8 game, you will know whether you have made your hand or not. Everything that you had to deal with before regarding probability and pot odds is no longer relevant. What is important is being able to read your opponents and know how your hand will fare against them.
A key to reading your opponents is to know when they bet. Understanding this will be able to help you narrow the range of hands they have. Do they bet lows? If so, do they only bet nut lows or do they bet 2nd and 3rd nut lows as well? What hands do they raise with? Only the nuts or is their raising range much wider? Do they check raise on the river and if so with what hands?
Let's say that you have A-3-4-5 and the board is 2-7-K-Q-8. There are five players in the hand and the big blind has bet every street and has again led out on the river. What you should do on the river will depend on the range of hands you believe the big blind to be betting with.
The first thing you will need to consider is your actual hand. Where do you stand with it? Do you have the nuts, something near it, or have you missed completely? When you hold both the nut low and a very strong high hand that is most likely good, your action is usually defined for you. Bet or raise until you can't any longer. The only time you should not bet is when you are positive (and by positive, we mean guaranteed) that your opponent(s) are going to do the betting for you. If in doubt – bet. In the above example, there are five people in the pot with you. You're definitely going to get called by at least one or two of them and some foolish player might even raise with what they believe is the best hand.
If you just have the nut low, however, then what you do will depend on your opponents and how many of them there are. Do you think anyone else has the nut low or are they betting/playing a high hand or non-nut low? The river card will help determine this. If it is a card that makes a high hand like a straight, full house, or flush, and your opponent who has been check-calling all of a sudden bets, it's likely they hit a high hand. Another time to bet a nut low is when the river card is one that counterfeited the previous nut low. For example, say you have A-2-4-x and the board comes 3-7-8-Q-2. Someone who had A-2 had their nut low counterfeited on the river and the probability that yours is the only nut low is much higher. In these instances, you are correct to bet and/or raise.
With the nut high hand, it is always correct to bet and/or raise. While it is true that you will occasionally split the pot with someone holding the same high, over the long run the odds are in your favor that you are winning half the pot. You can't get value from those 2nd and 3rd best high hands if you are just checking and calling.