First off I'm looking at Michigan law,
Sections 338.1066 and .1069 appear to make it illegal to wear or use a badge that identifies you as a security officer unless you're licensed and defined as a security officer under 338.1054. 338.1069 appears to restrict security uniforms as well.
If you are being employed or are exercising or representing yourself as a security officer you need to be licensed. Now part of this is inference as I have not read through every section, in Washington for instance one does not need a license if they're operating in-house security, meaning that say you have an HOA and they employ you to watch that HOA's property then you may not need one, you should contact your local police to see if that's the case.
However if there is no HOA you're arguably in the business of protecting other's property meaning you're contract and you would definitely need a license under this statue.
now if you are just an average joe involved in neighborhood watch for no money, I don't see any law that would require you to be licensed, but wearing security apparel and badge seems to be a definite no-no in that case.
Otherwise if you're just walking around your block you should have all your rights to openly or conceal carry a weapon in accordance with the law. I can find no michigan statutes on handcuffs being restricted and you appear to have some basic arrest authority
764.16 Arrest by private person; situations.
Sec. 16.
A private person may make an arrest—in the following situations:
(a) For a felony committed in the private person's presence.
(b) If the person to be arrested has committed a felony although not in the private person's presence.
(c) If the private person is summoned by a peace officer to assist the officer in making an arrest.
(d) If the private person is a merchant, an agent of a merchant, an employee of a merchant, or an independent contractor providing security for a merchant of a store and has reasonable cause to believe that the person to be arrested has violated section 356c or 356d of the Michigan penal code, Act No. 328 of the Public Acts of 1931, being sections 750.356c and 750.356d of the Michigan Compiled Laws, in that store, regardless of whether the violation was committed in the presence of the private person.
However read carefully you can only arrest for a felony, so you should become aware of what relevant offenses are felonies, also I don't know what other sections of michigan regulate that.
You really need to contact the state licensing authorities for security guards if you plan to work or act like one for your nieghborhood.