Can a landlord legally spy on their tentants? Or Have Neighbors constantly spy for landlord
This hinges on what you mean by "spy".
Generally, a landlord cannot enter a leased or rented property* without the tenant's consent, nor can their agents. (They can arrive and ask to enter, as can your neighbors whether or not you own your home, but you are not required to acquiesce in either case).
A landlord can view the publicly viewable portions of the property at their leisure, as can their agents, or any member of the public for that matter.
A landlord could possibly be notified of a tenant's actions in a number of ways: such as viewing the public portions of the property, being notified (or billed) by utilities or public agencies, or receiving complaints from the neighbors.
A neighbor has no more, and no less, legal ability to spy on you if you owned your home vs if you rent your home. So, they would have no more right to, say, spy at your house with a telescope than if you owned the property yourself, but no less right to complain if you have a loud (or audible) party or a large number of guests; the only difference being they can complain to someone who could potentially do much more than they could if you owned the property yourself.
Thus, the answer to your question depends on what is meant by "spying".
*This assumes that this is a separate property; a landlord who rents out a room in their own home often has far greater rights.