Are they supposed to send the patient home aspirating?

No, they should get a swallowing assessment done. This can be carried out by any qualified practitioner who has undertaken the appropriate training, but the real go-to people are Speech and Language Therapists whose knowledge of swallowing synchronicity and level of practice in listening to it are second to none.

When you say the patient "starts aspirating" do you actually mean aspirating or do you mean coughing and spluttering? Coughing means the reflex is intact. It's patients who don't cough who present the bigger problem.

So - no hospital admission, no feeding tube; what measures did the NH take?

Ask the questions GA suggested. From what I understand changing the consistency of the food is done. I would think the NH will try different things before they send the person back to the hospital. A feeding tube would be the last resort and I would think before I allowed it. The age of the patient and what health factors they have should be considered, Hospice may be the better option.

Was there a speech pathologist at the nursing home?   I'm assuming there was b/c someone apparently diagnosed aspiration.  

When my father had to be intubated, it was b/c he was unable to maintain a SAT rate above the then medical threshold level  (I think that he couldn't SAT at a much higher rate than the low 80s, but this was almost 20 years ago and I don't remember everything.)

Was her diet changed?   Was a videoscopic swallow test done?

I would think that, if there was a speech pathologist on staff, he/she would address the issue with a change of diet.  I'm not sure and rather doubt though that someone at a nursing home should insert a feeding tube; this is a hospital situation. 

Did she have aspiration pneumonia?    Where is she  now?