Drivers don't have issues with an uninvited dog passenger? As a dog owner, I find this surprising. But if drivers are OK with it it would be a great option for me because I don't own a car.
It depends on the city, culturally, and it's a good idea to call and ask first.
Obviously if it's a service dog they're required to transport it, but from knowing people with service dogs, it's still worth calling because some drivers are allergic and would rather you call another
Most drivers are okay if it seems like you have your pet neat. I take my cats in carriers all the time, but would never take my parents super sheddy dog.