How to consider two potential Product Owners?

I would go with the experienced PO and make the person proposed by the business as a key stakeholder/subject matter expert that the team can talk to whenever they have questions about the processes, etc. 

A better option would be to add that person to the team itself. 

Have either of these people been asked if they wish to be Product Owner, or had the opportunity to discuss the matter between themselves?

How would you handle this if both candidates were external hires?  Why wouldn't you do the same in this case?

You want the person that will best fill Scrum's role of Product Owner.  Interview them and let them convince you that they are the best person. Also review what the Product Owner role is responsible for and make sure that you are getting the right person.  https://scrumguides.org/scrum-guide.html#product-owner

And I like @Ian Mitchell's statement.  You just stated that two groups want to propose people but didn't indicate if either one of them applied.  

I'll start by saying that there's no one right answer. Depending on your exact situation, either could be the right thing to do. I'd start by echoing both Ian and Daniel - do both people understand what the role of the Product Owner is, what it entails, and want the role? If only one of the people wants the role, then you have your answer. If both want the role, only then do you have to make a decision.

I'd focus on the type of system that you are building. Are you making an in-house custom solution or are you trying to build something for the marketplace where you'll have to deal with competing stakeholder desires? If you're building a custom solution for the company's internal use and have no plans to market it outside, I'd lean toward the person coming from the business side. As long as the developers are strong technically and can provide the perspective on technical dependencies and input into how technical work impacts the ordering of the work, then having strong domain expertise will be immensely useful. However, if you're trying to build something for the general marketplace, then having more of a general product management background may be useful to help balance the needs of competing stakeholders and make the hard decisions, without having a lot of focus on the company's (who would possibly be one set of end-users) experience.

I do think that it would be useful to get the team's experience. Just because I would have a preference on the skill set, I'd also want someone that the team feels comfortable working with. The Product Owner will be working with the team every single day. It doesn't matter how much domain expertise they have or how much of a background in product management they have if the rest of the team doesn't feel they can work well together. The amount of good rapport needed from the beginning does depend on the context and how much time the team has to start to ramp up as a cohesive unit.

There's not going to be a one-size-fits-all solution here. Ideally, both people would continue to be involved in the team as stakeholders. It's a matter of who is the best to work with the rest of the team to ensure that the responsibilities of the Product Owner role are satisfied.

The best Product Managers I've worked with had an engineering background. In other words, people who can build software are the best to lead it. Business owners can work in a business architect or advisory role.

Which of the two will be given complete authority over all decision-making, ownership of the budget, and take full accountability for product ownership and valuable outcomes?

Hello,

I'm working in a product-based company. And really it's a very big product. So, in that situation, my PO should be a domain expert/SME and an experienced person. He should know that how to generate business value, how to make a decision if the complex conflict occures in a product, If developers stuck somewhere then PO should be able to give answers easily and take decisions immediately.  

So, I think, there is no single answer for this question. You have to think of different criteria to decide PO. 

How can we consider both without giving them 'only' a stakeholder role?

It may be possible for either or both of these people to join the Scrum Team as a Developer.

Perhaps these individuals have skills that would benefit the Scrum Team; just like it's often important to have business analysis, user experience or project management skills within a Scrum Team.

If the Scrum Team understand the stakeholders well enough, that may help them identify the main risks, and plan their work (or way of working) so they can learn the important lessons faster. If one of the Developers has product ownership skills, this may empower better collaboration between the PO and Developers.