Volunteers don't get paid.

Yes the value of the house should be reported on a 1099-NEC or W-2 if it was for services rendered.

The 990 reports it as wages or the specific category of service expense.

They want to give her a 1099S. That is not right because they did not sell a house. If the house is worth 400000. She would have to claim 400000 as earned income and pay self employment tax on it. Could they give her a 1099 and she could just claim as an asset.

"Could they give her a 1099 and she could just claim as an asset"

I don't know what this means.

Why are they "giving" her a house?

They are giving her the house as a gift.

Non profit wants to give her a 1099 so they can allocate the depreciation of the house.

It's a gift if it's a turkey at Thanksgiving.  Not a house.  Given in connection for her services, so the value of the property is compensation. 

There is leeway to regard this as 'other income' since she's not in the trade or profession of providing her services to the charity (admin work for example) and then not subject to s/e tax.  Likely disagreement on this point, but I'd disclose the position taken and put as not subject to s/e tax.

Something odd if the charity is giving her a house.  What do you mean by 'allocate the depreciation?'   Did a donor give the house to the charity, or does the charity run it for rental purposes such as parents needing to stay near hospitalized children?  Why did the charity give it to her?  How many years did she volunteer for them to have them do this?  Unusual I'd say!

"There is leeway to regard this as 'other income' since she's not in the trade or profession of providing her services to the charity (admin work for example) and then not subject to s/e tax."

But hold the phone.  How do you know that?  Maybe the house lady has a consulting business and part of that consulting is making nonprofits so happy that they give away $400,000 houses.