Many humans have created close relationships with wildlife and companion species. Notwithstanding that companion species were at some point themselves wild, some wild (i.e., feral) and domesticated (owned) dogs and cats now have significant impacts on wildlife. Many strategies exist to control the impact of dogs and cats on wildlife, but the successful implementation of management initiatives is tied to public opinions and the degree of acceptability of these measures. This paper reports the findings of a survey assessing the beliefs of residents in Queensland, Australia, about dog and cat impacts on wildlife, and their attitudes towards various strategies and options for controlling wild (i.e., feral) and domesticated (owned) dogs and cats. The responses of 590 participants were analysed. Our respondents collectively grouped strategies into those that directly cause wild dog and cat deaths and those that allow wild dogs and cats to live a 'natural' life, which is a variation on past research where respondents grouped strategies into lethal and non-lethal methods. Community acceptability of strategies that directly cause wild dog and cat deaths (each assessed using five-category Likert scores) was lower amongst females and respondents aged 34 years or less. Gender expectations in most places and cultures still predominately suggest that women are more 'caring', supportive of animal welfare, and perhaps cognizant that wild dogs and cats are also sentient creatures and appreciate the problematic tension between controlling wild and companion species. Age-related differences may reflect the changing social values of communities at different points in time. There was high support for regulations that enforce responsible pet ownership but not for the importance of pet-free suburbs, which the majority of respondents considered unimportant. These important variations in beliefs and attitudes require careful management within each community for the success of any program to control wild dogs or cats.