The answer is sometimes YES, vaccinated people are sometimes afraid of unvaccinated people, often with good reason. This in no way suggests the vaccine is not effective if potentially being exposed to the disease still concerns the parents of vaccinated children.
I blended my two comments into an answer.
Dr. Jodi Halpern wrote (from Ask the Experts: Should Schools Require Vaccines?).
Most parents of unvaccinated children believe they are trying to protect their child from vaccines' possible harm, despite medical evidence to the contrary. I've talked to parents who are terrified that vaccines will worsen the neuropsychological conditions of children with existing problems. They don’t want to put other children at risk. Rather, they are saying, 'We have a different belief system than the state, and the state cannot impose its beliefs on our family.'
Yet there is no doubt that an unvaccinated child infected with measles could inadvertently harm other children, especially ones who can't be immunized because they have low immunity due to illnesses like cancer. So there’s an ethical and fiduciary responsibility that lies with the school.
There are basic conditions to have a civil society, and one of them is collaboration. For parents to send their children to school, they have to believe that the schools are full fiduciaries for their children. They have to trust that the school will not harm their children.
You wouldn't send your child to a school with broken glass, rusty nails, or dangerous equipment on the playground. You expect the school to keep its grounds safe. In the same way, when you send your young children to a public school, you expect that the school will protect them from preventable, serious diseases. If your children are going to be put at risk for a serious disease like measles, then the schools are not being an adequate fiduciary.
For this reason, public schools do have an ethical obligation to require students to get vaccinated unless they are immunocompromised and can't for medical reasons. This does not mean that the state has the right to forcefully vaccinate children against their parents' will. But it does mean that some children whose parents don't believe in vaccines may not be able to attend public schools during measles outbreaks if their presence would put other students at risk.
I think above is the basis of the answer. Yes, some parents of vaccinated children do fear it as their other children may die. Now anti-vax groups may spin this as evidence that people with vaccines are worried they may not work! This is spin. A small group of parents of vaccinated children fear it with very good reason. One of their children may die.
So you have two children: (A) one healthy 10 year old child vaccinated, and (B) one 12 year old child with a serious illness that would result in death from a common disease we vaccinate against (as stated in the above quote) but who cannot be vaccinated due to their illness. Do you want (A) hanging out with an unvaccinated child? Fear = Yes, as (B) could die! Let's say it is only 1% chance of your vaccinated child bringing home the disease. Do you want to take that 1% with your other child's life! See @Roger comment with the bulletproof proof vest analogy. Vaccines are not 100% we rely on the herd immunity that these unvaccinated children can destroy.
Additionally, I think many parents would not want to associate with these children's parents (parent to parent not child to child as your questions asks). I would consider someone who holds back vaccines for no proven medical reason as ignorant (similar to a racist or bigot) so I would not want to associate with them and I would not want my child to overly associate with their child, say at a sleepover as their parents have already demonstrated very poor decision making. This could be extrapolated into, "parents with vaccinated children don't want them to hang around unvaccinated children" but it is not fear.
Some additional resources to back up my answer.
"Why I won't let unvaccinated people around my kids", CNN
"To the Parent of the Unvaccinated Child Who Exposed My Family to Measles", Mother Jones (--Just a note before reading this the story is quite harrowing).
Just of the top of my head I can come up with at least 10 reasons I would fear my child hanging out with an unvaccinated child. For brevity I will not list them all but one is if any member of my household worked on a children's cancer ward.
*Please note: The bold items are to highlight the specific related claims are not highlighted in the original quoted text.*EDIT - I am going to update the answer as the question has changed a little.
In 2000, "The US Centers for Disease Control and Prevention (CDC) declared endemic measles eliminated in the USA" according to Gostin, L. O. (2015). Law, Ethics, and Public Health in the Vaccination Debates: Politics of the Measles Outbreak. JAMA, 313(11), 1099-1100. Chicago
However, vaccine-preventable diseases (eg, diphtheria, pertussis,
tetanus, measles, mumps, and rubella) are increasing, with some
parents delaying or selectively immunizing their children and, at
times, even opting out of having their child immunized. In 2014, the
United States recorded a record number of measles cases—644 cases from
27 states, more than 3-fold higher than any previous year since 2000.
As of February 6, 2015, the CDC has reported 121 measles cases in 17
states, mostly from an ongoing outbreak linked to an amusement park in
Orange County, California. Most cases were unvaccinated (55%) or of
unknown vaccination status (31%).
This quote is interesting pertaining to the original question.
“Their children have been sent home from school. Their families are barred from birthday parties and neighborhood play dates. Online, people call them negligent and criminal.” (Healy J, Paulson M. Vaccine critics turn defensive over measles. New York Times. January 31, 2015:A1.)
Finally, evidence to the contract often forwarded by the antivacs has been debunked.
Reiss, Dorit Rubinstein. "Herd Immunity and Immunization Policy: The Value of Community Protection." Oregon Law Review 94 (2015). whi state,
This article explains why claims made by Holland and Zachary in their
article, Herd Immunity and Compulsory Childhood Vaccination: Does the
Theory Justify the Law?, are incorrect and untenable. Their view of
herd immunity is also incorrect: the article does not well define the
term, ignores data showing that herd immunity works, and their
discussion of their two examples is inaccurate: close examination of
those examples actually shows the role of herd immunity in protecting
I should note the question has changed since I posted this answer so it may not seem to be a perfect 'fit' now but it was when the question was asked. I hate it when this happens.