can we declare local inner class as private?

No, local inner class cannot be declared as private or protected or public because local inner class is not associated with Object.

Example of local inner class

public class Main{ 
     private String website="826.a00.myftpupload.com";
     void show(){  
          class LocalInnerClass{  
               void display(){
                   System.out.println(website);
               }  
          }  
          LocalInnerClass localInnerClass = new LocalInnerClass();  
          localInnerClass.display();  
     }  
     public static void main(String args[]){  
      Main obj=new Main();  
      obj.show();  
     }  
}

Output

w3schools.com

Above of local inner class with public access specifier

public class Main{ 
     private String website="826.a00.myftpupload.com";
     void show(){  
          public class LocalInnerClass{  
               void display(){
                   System.out.println(website);
               }  
          }  
          LocalInnerClass localInnerClass = new LocalInnerClass();  
          localInnerClass.display();  
     }  
     public static void main(String args[]){  
      Main obj=new Main();  
      obj.show();  
     }  
}

Output

Main.java:4: error: illegal start of expression
          public class LocalInnerClass{  
          ^
1 error

Note: Same case will be for private and protected access specifiers.