I did run some quick (successful) tests with paper-input. Keep in mind that your paper-inputs must have a "name" attribute in order to be submitted along with other form fields inside of an ajax-form. This requirement is consistent with the way a native form treats fields that lack a name attribute (it ignores them).
ajax-form 1.0.0 will "submit" any custom element in the form if it contains a name attribute and has a value. The next step is to respect any constraints attached to a custom form field element. core/paper-input exposes an "invalid" property to determine if the field is valid. In v 1.1.0, ajax-form will likely check this property on all custom form fields (in addition to validation checks iaja-form already runs on native form fields) to determine if the form can be "submitted". It seems reasonable to carry this convention forward and expect all future custom form fields to contain an "invalid" property to make this determination as well.