From the Guidelines

Isolation is necessary for shingles until the rash has crusted over, as recommended by the Advisory Committee on Immunization Practices (ACIP) 1.

People with active shingles should avoid contact with individuals who have never had chickenpox, especially pregnant women, newborns, and those with weakened immune systems.
In healthcare settings, standard precautions plus contact precautions are typically implemented, and airborne precautions should be employed for patients with varicella or disseminated herpes zoster (HZ) 1.
The person with shingles should keep the rash covered, avoid scratching, practice good hand hygiene, and avoid sharing towels or clothing.

Isolation should continue until all lesions are dry and crusted, which typically takes 7-10 days.
For immunocompetent persons with localized HZ, standard precautions and complete covering of the lesions are recommended until the lesions are dry and crusted 1.

Shingles is caused by the reactivation of the varicella-zoster virus, and can be transmitted through direct contact with the fluid from shingles blisters.
While someone cannot catch shingles from another person, they can develop chickenpox if they've never had it or been vaccinated against it.

Airborne precautions, such as negative air-flow rooms, should be employed for all patients with varicella or disseminated HZ, and for immunocompromised patients with localized HZ until disseminated infection is ruled out 1.
If negative air-flow rooms are not available, patients should be isolated in closed rooms and should not have contact with persons without evidence of immunity to varicella.

From the Research

The need for isolation for shingles is not explicitly stated in the provided studies, but measures for preventing the spread of the varicella-zoster virus include infection control through routine hand hygiene and appropriate use of isolation precautions and personal protective equipment 2.
Herpes zoster is caused by reactivation of the varicella-zoster virus, which also causes chickenpox, and patients with conditions that decrease cell-mediated immunity are more likely to develop herpes zoster 3.
The varicella-zoster virus vaccine is effective in reducing the incidence of herpes zoster and postherpetic neuralgia, and is recommended for adults 50 years and older 3, 2.
Treatment of herpes zoster includes antiviral agents, such as valacyclovir or famciclovir, which should be started within 72 hours of onset to reduce the severity of the infection and the duration of the eruptive phase 3, 2, 4.
While isolation is not specifically mentioned as a necessary measure, it is implied that preventing the spread of the virus is important, especially for immunocompromised individuals who are at higher risk for complications and recurrence of herpes zoster 2.