For starters, there is more than one definition of transitivity out there. The German Wikipedia points out that traditional (German) school grammar only considered verbs with an accusative object transitive, so etw.{akk} essen would be considered transitive but jdm. helfen would not. It also mentions that modern linguistics consider transitivity in a more broader sense as highlighted below.[citation needed]
Canoo.net also follows the idea of transitive verbs being those with an accusative object but also constructs an additional distinction between ‘true’ transitive verbs that can form a passive voice and ‘pseudotransitive’ verbs that cannot form a passive. An example for a pseudotransitive verb would be haben: ‘Das Buch wird gehabt’ does not make sense.
christianlehmann.eu uses the formal definition of a direct object that the English Wikipedia also uses. He notes that in German, being able to form a passive voice (with a subject) is sufficient and necessary for a direct object while in English it is merely necessary.
Note that the introductory statement of the German Wikipedia article on transitivity is that a verb is transitive if it takes exactly two arguments,[1] so jdm. helfen would be transitive despite that forming a passive without a subject.
All sources more or less agree that there are verbs that can be used both transitively and intransitively, essen or trinken being examples of those. However note that the meaning is a tad different in each case and some languages may use different verbs depending on whether the transitive usage or the intransitive is required: ich esse just means I am shoving food into my mouth while ich esse das Brot concentrates more on the bread and not on the process of eating.
Now depending on the definition, the only verbs that are clear are those that clearly require at least one (direct) object and thus are clearly transitive. *Ich erschieße is an invalid sentence and thus erschießen must be transitive. So there is no easy rule around.
Still, for many verbs one type of usage is overwhelmingly common thus leading to the other usage being almost forgotten.
[1]: Arguments are add-ons for verbs that cannot be omitted without drastically changing the meaning. Many verbs in passive voice are essentially zero-argument:
[…], dass gearbeitet wurde.
The sentence works without anything, there is no subject and no object. Verbs with one mandatory argument include leben:
Ich lebe.
And finally, two or more arguments would always include a subject and at least one object:
Ich helfe dir.
Ich mag dich.
Ich stelle dich ihm vor.