Some sources claim that breathing into a paper bag is a good way to control hyperventilation caused by a panic attack. The theory (I surmise) is that re-breathing CO2 helps to mitigate the respiratory alkalosis caused by hyperventilation. If the alkalosis is perpetuating the anxiety, this seems like a logical intervention. Has this been shown to be an effective intervention for panic attacks?
I did not find any studies which would investigate that matter. Moreover, in 2008 Nillni and co-workers (1) stated:
Surprisingly, although the expressed goal of breathing training is to correct hyperventilation, pCO2 has never been used as an outcome measure
In a recent study (2) it was concluded:
Clinical improvement must have depended on elements common to both breathing therapies rather than on the effect of the therapies themselves on CO(2) levels.
As so I would conclude that even in theoretical level there is very limited evidence for that intervention to be effective.
But I would your use very different perspective. I bet that if you choose ten people out of the street and ask them "What would you do if your friend had a panic attack?" more than half would respond that they would give him/her a paper bag. The use of paper bag as an emergency intervention is a very common factoid. With this matter there exists a major placebo effect with that treatment. Use of paper bag as an emergency measure for person with a panic attack is very cost-effective without any potential side effects so it is an ideal "intervention". So I think we should not crave for solid evidence. Due to the factoid matter majority of people think it is an effective treatment and with comes a great placebo effect which should be definitely used due to the cost-effectiveness and safety.
Breathing into a paper bag to control hyperventilation might improve hypocarbia, but should NOT be done because of the risks of hypoxia and death.
This case report shows that breathing into a paper bag during hyperventilation can reduce partial pressure of oxygen by up to 42 mmHg (normal 80-100mmHg) and can result in death.
As answered by @arkiaamu, a better treatment would be reassurance, pursed lip breathing, and breathing exercises to restore regular breathing pattern.
Case Report Source: Callaham M. Hypoxic hazards of traditional paper bag rebreathing in hyperventilating patients. Ann Emerg Med. 1989;18(6):622-628. doi:10.1016/s0196-0644(89)80515-3