AviK
US Embassies and consulates abroad are NOT US Soil. You might want to read the Vienna Convention on Diplomatic Relations (1961) Articles 22 and 23.
The Vienna Convention sets out the rules for Embassies and the US is a party to it.
Article 22
1. The premises of the mission shall be inviolable. The agents of the receiving State may not enter them, except with the consent of the head of the mission.
2. The receiving State is under a special duty to take all appropriate steps to protect the premises of the mission against any intrusion or damage and to prevent any disturbance of the peace of the mission or impairment of its dignity.
3. The premises of the mission, their furnishings and other property thereon and the means of transport of the mission shall be immune from search, requisition, attachment or execution.
Article 23
1. The sending State and the head of the mission shall be exempt from all national, regional or municipal dues and taxes in respect of the premises of the mission, whether owned or leased, other than such as represent payment for specific services rendered.
There are a few phrases, all in Article 22, which have to do with mission sovereignty.
1. The premises of a mission shall be inviolable
Nobody can enter the mission without permission- this includes the host country
2. The receiving State is under a special duty to take all appropriate steps to protect the premises of the mission against any intrusion or damage and to prevent any disturbance of the peace of the mission or impairment of its dignity
If you visit the US Embassy, especially in more dangerous parts of the world, you will see the exterior of the mission being guarded by local nationals. The inside will have a marine detachment, but the outside will either be local police or military.
Similar to number one, unless the individual or party is asked to conduct any of the above, it is not allowed by a host country or any entity. The premise, just like the diplomat and their family, is awarded immunity.
If you are curious about what diplomatic immunity entails, you should read Articles 29-41.
Though the immunity can be violated, for the most part it is respected. Without such immunity, the life of a diplomatic would be more difficult during times of aggression within and between countries.