This is one of the most searched questions in the essential oil category, and it deserves a direct answer, however, you should always consult your vet and this is not medical advice.
Here is what the evidence says.
Lavender and dogs
Diluted lavender essential oil is generally considered low-risk for healthy adult dogs when used appropriately. The key word is diluted. Pure, undiluted lavender essential oil applied directly to a dog's skin can cause irritation. It should not be given orally.
Used as a passive room scent on a stone diffuser at normal levels (a few drops in a personal space), lavender essential oil is unlikely to cause harm to a healthy adult dog. Dogs have a significantly stronger sense of smell than humans, so strong concentrations can be overwhelming and stressful even if they are not toxic.
Watch for signs that your dog is bothered by the scent: leaving the room, pawing at the nose, excessive drooling, or watery eyes. If you see any of these, air out the space and discontinue use. A dog that does not want to be in the room with a scent should not be made to stay.
If your dog has respiratory issues, is very young, or very old, talk to your vet before using any essential oils in your home.
Lavender and cats
This is where the answer changes.
Cats are significantly more sensitive to essential oils than dogs. Cats lack a liver enzyme called glucuronyl transferase that helps metabolize many plant compounds, including some found in essential oils. This means compounds that a dog or human processes without difficulty can accumulate to toxic levels in a cat.
Lavender contains linalool and linalyl acetate, two compounds that cats cannot metabolize efficiently. The ASPCA lists lavender as toxic to cats. Symptoms of lavender toxicity in cats include vomiting, lethargy, loss of appetite, and in severe cases, respiratory distress.
This does not mean a whiff of lavender will harm your cat, and brief, incidental exposure to a low concentration is unlikely to cause acute toxicity. The concern is chronic low-level exposure from diffusers running in spaces where cats spend a lot of time.
If you have cats and want to use essential oils in your home, keep diffusion to well-ventilated spaces your cats do not occupy, keep diffusers out of reach, and watch for any behavioral or physical changes. When in doubt, do not use essential oils around cats and talk to your vet.
Why the diffuser type matters for pets
This is where stone diffusers have a real safety advantage over ultrasonic diffusers for pet owners.
An ultrasonic diffuser creates a mist that carries oil droplets into the air. Those droplets can land on surfaces, on your pet's fur, and on their bedding. A pet that grooms itself may then ingest oil residue from their coat.
A stone diffuser does not produce mist. The oil evaporates directly from the stone as a vapor, not as droplets. This means nothing lands on surfaces or on your pet's fur. The exposure is purely through the air and at lower concentrations than a misting ultrasonic.
If you have pets and want to use essential oils, a stone diffuser in a well-ventilated space where your pet can come and go freely is the lower-risk option.
Also: no reservoir. No standing water for a curious pet to tip over or drink from. No cord at pet level to chew through.
The main preventative measure for the stone diffuser is to keep it out of your pets reach so they do not lick the stones with the essential oil on them, you want to avoid ingestion.
Other essential oils and pets
Lavender gets most of the attention, but it is not the only oil worth thinking about when you have animals.
Avoid around cats: tea tree (highly toxic), eucalyptus, peppermint, citrus oils, cinnamon, clove, thyme, oregano.
Avoid around dogs: tea tree at any concentration, pennyroyal, wintergreen, ylang ylang, cinnamon.
Generally lower risk for dogs at low concentrations in well-ventilated spaces: lavender, frankincense, cedarwood, chamomile.
This list is not exhaustive and is not veterinary advice. If your pet has an existing health condition, is pregnant, nursing, very young, or very old, the bar for caution should be higher.
What to do if you think your pet was exposed
If your pet shows any symptoms after being in a space with essential oils, contact your veterinarian or the ASPCA Animal Poison Control Center (888-426-4435). Keep the oil bottle available so you can tell them exactly what the animal was exposed to.
Practical guidance
Keep all essential oils out of reach of your pets to avoid ingestion. Use a stone diffuser rather than an ultrasonic, and keep it in a room with good ventilation. Do not diffuse in an enclosed space where your pet has no way to leave. Start with low amounts and watch how your pet responds, and if they leave the room, let them.
For cats specifically: keep diffusion minimal, in rooms your cats do not sleep in, and consult your vet if you have any questions about their specific situation.
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Related reading:
Stone diffusers vs. ultrasonic diffusers: an honest comparison
The difference between essential oil and fragrance oil
How to read an essential oil label