Yes, you can easily hear “personne me connaît” instead of “personne ne me connaît” in spoken french.

I think this is true with any ‘negating’ structure (“ne … jamais”, etc.), as long as the second item convey a negating meaning in itself.

Indeed, “ne” can always be omitted in spoken French (“ne … que“, “ne … jamais“, …).

If you are trying to speak or write in a formal way however, I would recommend never to omit it in an official document¹.

When spoken, it is also usual to shorten it, just pronouncing the N (a sort of a contraction). When speaking formally, “Je n’suis pas d’accord.” will be better than “Je suis pas d’accord.“

Additional note: in literary French, “pas” can be omitted in “ne … pas“. This is very rarely heard in spoken French however, and is specific to “ne … pas“.

_{1. My first instinct would be to compare it to the inflected negative auxiliary verbs (e.g. the “don’t” contracted form) in English, which I would avoid in formal documents. Unlike the French omission though, the contraction is not considered unformal according to American recommendations (which can be summed up as “write as you talk”) and English ones (thanks to mikeyreilly for pointing that out).}

If Personne might have the meaning of English “nobody”, I would like say yes, the ne could be omitted in spoken language.

Since I’m not native speaker in French, the personne word alone feels not enough to me to express (at least, not with the same feeling strength that the English word “nobody” transmits to me), then I couldn’t omit the ne from it.

However, as said before, I would recommend never omit such words — even it looks a bit useless at a glance. Also, it reminds me that sentence in which they omitted the pas: je ne suis Charlie.

Etymology

Wiktionary tell us that :

(Nom) Du latin passus (« pas, enjambée, marche » ou « trace de pas »).

(Adverbe) De l’usage en ancien français d’ajouter un substantif signifiant « le moindre » après ne :

Je ne bois goutte. — Je ne mange mie. — Je ne marche pas.

A “pas” is a step, the shortest measure of distance. In old french, to negate, we were using “ne” with the shortest measure of what we were doing.

Je ne bois goutte

→ I do not drink, even a single drop.

Je ne mange mie.

→ I do not eat, even a little piece of bread (“mie” is the inside of the bread, but underline the special meaning here)

Je ne marche pas.

→ I do not walk, even a single step.

L’usage de pas s’est généralisé par un cycle de Jespersen et la négation ne est pour sa part devenue optionnelle dans le langage courant :

Je marche pas dans cette combine. — J’vois pas où tu veux en venir.

_{I do not walk in this ploy (I don’t accept). — I do not see where you want to go (what you mean).}

So, in conclusion, we can use “pas” for nearly every negation and “ne” became optional progressively.

General case

To come back to what you are asking :

However, I wonder if “ne” can also be omitted in other structures, notably with “personne”.

Yes, you can in all cases when you are talking. When you are writing or in more formal context, it is necessary to write/say the full negation.

As @Chop says :

When spoken, it is also usual to shorten it, just pronouncing the N (a sort of a contraction). When speaking formally, “Je n’suis pas d’accord.” will be better than “Je suis pas d’accord.”

You example :

For example, “Personne ne me connaît ici” means that “No one knows me here”, but can I speak “Personne me connaît ici” to express the same thing?

“Personne me connait ici” will be understood as “Nobody knows me here”

It is very important to speak properly before attempting to sound casual or informal.

“Personne me connait” is a mistake. There is no doubt about it.
It is tolerated when fluent french speakers are talking to friends in informal situations.

It is otherwise perceived as a grammatical mistake.

“Personne,” without the negative particle “ne” would be regular in speech in brief answers:
“Qui a fait ça?” “Personne.”