Activity trackers for dogs are increasingly popular, having the potential to improve pets' welfare and providing a 'digital voice' for expressing their needs. ACI research has so far mainly focused on their impact on the pet-human bond. However, also privacy considerations play an important role as they may pose significant barriers towards their wider adoption. We report on a mixed-method study (N=61) investigating what, if any, privacy concerns dog owners hold towards the data captured by their dog's device. We elicited detailed reflections by participants towards the consequences for themselves and others of a hypothetical data breach leaking their dog's data. In addition, we captured several potential indicators for the perception of consequences: trust, perceived transparency, risk, benefit, and self-assessed knowledge of dog behavior (and thus its data). Statistical analysis of the findings indicated that perceived consequences were moderately correlated with trust and perceived benefit of use for society as a whole. A thematic analysis revealed that participants either did not see any consequences, saw consequences only when reasoning about others, or saw consequences to their own or dog's safety, rather than their privacy. We discuss why these findings are worrying in light of the information asymmetry between consumer and service provider, setting out an argument why dog owners should care more about dog activity data and its privacy implications due to the data's ability to reveal potentially sensitive data about themselves as well as their caregiving.