The problem with this is that there are multiple competing needs and desires here. People may have a desire for that personal space, but many will sacrifice those when it helps them meet other needs.
Yes, it's great to have personal space. But you know what? People sacrifice personal space for spending less money all the time. And for convenience.
Consider e.g. London - a city awash with money, where a lot of even people who could afford to be driven around by a personal chaffeur will opt for public transport for convenience.
As another example for London: People will opt to smell someones armpit on the train in the regular train carriages rather than pay a few pounds extra for a seat in the First Class carriages all the time. The value of personal space when travelling, as it turns out, is deemed by large parts of the public to be very low.
People tend to opt for personal space mainly when the public transit options are unusably bad compared to driving.
Places like London are perhaps the areas where this transition is most likely to start: Places where those who even own cars often own cars as a "contingency" for those times when the bus doesn't arrive or you're going somewhere odd that just doesn't work well with public transport currently.
In those cases, for a lot of people, it'd be very attractive to e.g. pay a membership fee to guarantee a certain level of "contention" for cars to be able to just press a button and have one arrive "fast enough". Even more so with the ability to do that on either end of a train ride. For a lot of people this will make a car pointless.
And in environments like this "personal space" is moot, as almost everyone are already used to using public transport some or most of the time.
At the same time, it is somewhere where local authorities are clamouring for ways to reduce parking and make car ownership less desirable. Expect housing units to start coming without parking spaces or with very few parking spaces in high density areas as cost saving measures, or because they'll sell some of those parking spaces to ride share companies, or make residents who want them buy them separately.
Expect planning rules to start reducing the maximum allowable number of parking spaces.
Places with plenty of space, sprawl and a strong culture of cars as independence will certainly experience this change last.
But consider e.g. the impact of a generation of youth who will eventually grow up with a situation where they may be able to rent a self-driving car on demand from before they are able to (afford to) buy one, and where e.g. parents may opt to just order a journey rather than driving them somewhere once old enough, and who will grow up increasingly likely to get used to some car taking them somewhere without needing to take the step to car ownership to get that freedom from parents driving them around. The "liberation" may become to be able to sign up for your own account so your parents can't see your every journey.
I think that the whole culture where car ownership is seen as a rite of passage and signifier of liberation from your parents could change far faster than you think.