Private betas can be great when you have lots of existing customers, and you'd like to limit the quantity of feedback on a new product or feature.
If this would be the first exposure the app is getting, and you don't have a treasure trove of users that might be lined up to get into the beta, I'd vote for a public beta.
In fact, I'd go further and not focus so much on the nature of the beta, but just launch the product, noting that this is the initial release, and you'd love to get feedback as you continually improve the product.
The three most important things for a startup to do are: Ship it, ship it and ship it. Location matters much less these days ;)