Just throwing a "mir" there does not really have the same strong emphasis on the person as "myself" in English. The "mir" there does just slightly change the meaning of the verb, it does not add anything else.
- Sie sollten es sich selbst anschauen (they should (personally) take a look)
- Sie sollten es sich anschauen (they should take a look)
So adding "selbst" is usually the way to have the same emphasis as "myself" as in English.
EDIT: There are however some cases where the english sentence does not really work without an object and then "sich" can become "myself/yourself/themselves/etc.".
- Ich muss mich anmelden (I have to register myself)
If you there would add "selbst" there, it would change it to:
- Ich muss mich selbst anmelden (I have to register myself myself ;) meaning: I have to do it, nobody can do it for myself)
"mir" is there because "anschauen" is a reflexive verb that takes a dative pronoun http://joycep.myweb.port.ac.uk/abinitio/chap10-10.html
When there is an object, it takes dative. Else, accusative.
Example: "Ich werde mir das anschauen" - das is the object.
"Ich werde mich eintragen" - no object.
"Ich wasche mir die Hände" - die Hände is the object.
"Ich ergebe mich" - no object.
"Ich wünsche mir ein Pony" - ein Pony is the object.
"Ich freue mich" - no object.
Hope you were able to understand my explanation. :)
Thank you for this extremely clear explanation. Regardless of whether it is a hard and fast rule or just a rule of thumb, it will definitely help.
Also, thank you for using the word "object" as opposed to quoting "the German xyz case". Using the terms that a native English speaker was taught while growing up makes these concepts so much easier to grasp. I would really appreciate it if more people here would follow your lead. Sir, you are a beacon of clarity in a forest of confusion.
"By myself" changes the meaning to the subject acting alone. The sentence doesnt mention "allein / selbst", the verb just happens to be reflexive. A good example os setzen bc it makes more sense in English: "I sit myself down". The "myself" is the way German reflexives work.
Rather than "I sit down by myself" which has a different meaning of doing so alone... hope this makes some sense. You just have to remember which verbs are reflexive in German. Some like anschauen dont make a lot of sense to an English speaker i.e. we would never say "I look myself at that".