How can one determine if the opposite of an agent noun exists?

We know that the employer employs the employee and that the tutor tutors the tutee, but how do we know if the shooter shot the shootee?

Is there a simple way to determine if an agent noun can be made into the object of the agent noun's action? In some cases, this is plainly obvious. We have runners but definitely not runnees and we have jumpers but not jumpees. There ought to be a way can we easily determine if the shooter shoots the shootee since a shootee is intuitively the person who gets shot by a shooter.

Note: Since the meaning of shootee can be intuitively determined by any English speaker, it isn't technically incorrect according to a descriptive linguist, but how would a prescriptive grammarian feel about it?

The -ee / -er distinction turns out to be a (rare) example of Ergativity in English.

As it says in the link,

Adding the suffix -ee to a verb produces a label for a person who is the Absolutive of the verb – i.e, a person who is either the Direct Object of a Transitive verb, or the Subject of an Intransitive verb.

Intransitives:

• Bill has retired → Bill is a retiree.
• Bill has escaped. → Bill is an escapee.
• Bill is standing. → Bill is a standee.

Transitives:

• They employ Mary → Mary is an employee.
• They inducted Mary. → Mary is an inductee.
• They appointed Mary. → Mary is an appointee.

Fun, huh?

There are two ‘rules’ here:

• one ‘rule’ is that a verb which takes a personal object—direct or indirect, but in some sense the recipient of an action—can produce a noun designating such a recipient by appending -ee (or -e if the verb already ends in e): shootee, tutee.

• another ‘rule’ is that a verb which denotes a change of personal state can produce a noun designating the person who suffers (or enjoys) that change with the same suffix: retiree, escapee.

These ‘rules’ will produce ‘well-formed’ words, which will certainly ‘exist’ in some sense; but whether use of these words will prove ‘acceptable’ within any specific speech community is another matter, for which there is no ‘rule’ but ‘Run it up the flagpole and see who salutes’.

Transitive or ditransitive verbs that can take a human direct object (or human indirect object, in the case of distransitives) are possible targets for the -ee suffix. The grammar of this suffix depends not just on the syntax but upon semantic criteria.

• If the verb is not transitive, there is no object and no -ee. EDIT: As @John Lawler points out, there are indeed intransitive usages like escapee.

• If the verb is intransitive and the subject is human, then -ee corresponds to the subject.

• If the verb is transitive but can't take a human object, then -ee is not appropriate.

• If the verb is ditransitive like give or tell, then the -ee corresponds to the indirect object.

In all the above cases, the -ee will be understood semantically, but it may not be considered proper or formal. E.g. givee is understood as the person being given, but recipient is preferred.