Allergy testing is a very effective method for controlling allergies; however, it can be expensive and requires either the pet owner or the veterinarian to give periodic injections.
You use 'either ... or' to refer to two items of which any one can be considered.
And you use 'neither ... nor' to link two things that are not true or do not happen. 'Neither Michael nor David wants to go to the picnic' means both of them do not want to go.
Note that when using 'neither ... nor', the verb generally agrees with the nearest noun: 'Neither my sisters nor John is keen on going to the picnic.'
You may use 'is' or 'are' in 'neither of us is/are going to the picnic'. A plural verb is usually used in everyday speech, though a singular verb is correct in formal language.