There are a few considerations to make regarding the thought process of "if the other kid is vaccinated, then they should stay healthy," even when exposed to illnesses that they are vaccinated for.

(1) Vaccines are not 100% effective. In the case of the Measles vaccine it is estimated to be 93-97% effective -- this is a very effective vaccine. So, if someone is vaccinated, then yes, they likely will stay healthy even if they are exposed to a case of the Measles.

(2) Not everyone can be vaccinated for medical reasons whether it is due to allergies to something in the vaccine or another medical issue. So, these people are forced to rely on what is called herd immunity (everyone who can be vaccinated around them is vaccinated, so the virus or bacteria will not be around to infect the unvaccinated person). Unfortunately, for herd immunity to work specifically for the Measles scientists believe that 95% of the population needs to be immune to stop its spread. This is because the Measles is extremely spreads extremely easily -- about 90% of people who are not immune to Measles will become ill when they are exposed to the Measles.

(3) The last point that I will make is that if a pathogen (virus or bacteria) is allowed to circulate in the population due to low vaccine uptake, then there is a stronger likelihood that the pathogen will mutate (change) to get around the protection of the vaccine. Then nobody is protected and scientists get the fun of trying to create a new vaccine for the mutated version.

Take what I say here with a grain of salt since I am not an immunologist, physician, or scientist. I just like to know the pros and cons of vaccines as well as how best to protect myself, my family, and my friends from preventable illnesses. I learned a lot of this information by talking to my doctor, reading from medical journals (Lancet, New England Journal of Medicine, Nature Immunology, etc.), and also checking major medical center internet sites for information such as Cleveland Clinic, Johns Hopkins, Mayo Clinic, MassGeneral, etc. There is a lot of excellent information to be had from our scientific community to help make an informed choice - much of what I located was open and free to the public to read.

Not all vaccines are 100% effective, and not everyone is able to get a vaccine (such as immune suppressed or immune compromised).

Even if the vaccine is only partially effective, it will reduce the viral load and speed up recovery time, which is very effective at reducing the number of people that a vaccinated person can spread their illness to.

To prevent an outbreak, an infected person must spread their disease to an average of less than 1 person. Having more people vaccinated is the easiest way to reduce this number.

An immune suppressed person can still be protected by having a majority of people vaccinated, and therefore unlikely to spread a disease to a person who interacts with them.