Should Lay Buddhists Teach the Dhamma?

Yes, if he as enough knowledge.

The required lowest graduation:

Qualities of enlightened lay teacher for lay people:

1. Enlightened sodāpatti-phala, at least.
2. Proficient to recite and to understand 9 pāṭha, the whole K.N. khuddaka-pātha, and their commentaries.
3. Proficient to recite and to understand the whole K.N. Itivuttaka, and their commentaries.
4. Enough understand to judge/to decide about 8 precepts.
5. Enough understand to judge/to decide about abhidhamma in his kammaṭṭhāna. This abhidhamma knowledge will come with his meditation and his enlightenment, no need to study. However, more abhidhamma knowledge is required for teaching the other qualities below, so lay teacher must study this part, such as Cittagahapati did.
6. 5 years experience in buddhism.

I sum above qualities myself from the profile of lay-etadagga, such as Khujjuttarā, Cittagahapati.
However, the pāli canon never focus on lay teacher, because the ariya-lays very respect bhikkhu. So, they often invite bhikkhu to teach, first. But khujjuttarā can not invite bhikkhu into the king's palace, so she must recite & recollect tipitaka herself to teach inside the king's palace.
Another, if lay teacher will teach bhikkhu, he need this Qualities, that was writen in commentary according to V.N. Mahākhandhaka:

Qualities of nissayamuccaka-bhikkhu (teaching lay people)

1. Proficient to recite pāṭimokkha-pāli and to understand it's commentary.
2. Proficient to recite and to understand 4 bhāṇavāra (~1,000 syllable) of sutta and their commentary, to teach laymen on uposatha day.
3. Proficient to recite and to understand sutta for bhikkhu's life such as andhakavindasutta, mahālahulovādasutta, ambaṭṭhasutta, etc.
4. Proficient to recite and to understand sutta for teaching in 3 chances: banquet for saṅgha by layman (nidhikaṇdasutta), funeral ceremony (tirokuṭṭasutta), and auspicious ceremony (maṅgalasutta).
5. Enough understand to judge/to decide about saṇgha's ceremony such as uposatha, pavāraṇā, etc.
6. Proficient to recite and to understand his kammaṭṭhānā throughout the nibbāna-course.
7. 5 years experience in monk hood as a monk.

Qualities of bhikkuparisūpaṭṭhāpaka-bhikkhu (teaching bikkhus)
If above layman's teachers want to teach bhikkhus (ūpajjhā-ācāriya, nissaya-ācāriya), they must increase their skill level to all of the following qualities.
These are for abhivinaya teaching:

1. Proficient to recite mahāvibhagha and bhikkhunivibhaṅga (first 3 books of thai 45 books pali-tipitaka) of vinaya-pitaka-pali. At least, he can relay with the other 3 bhikkhu. Proficient to understand it's commentary, too.
2. Proficient to recite all saṇgha's ceremony in vinaya-pitaka mahāvagga and julavagga.
3. Proficient to recite 14 vatta in vattakhandhaka.

These are for abhidhamma (kammaṭṭhāna) teaching:

4. Proficient to recite one of this suttanta-pali: mūlapaṇṇassa (1st/3 parts of M.N.) for student in M.N. faculty, mahāvagga (2nd/3 parts of D.N.) for student in D.N. faculty, sagāthavagga+nidānavagga+khandhavāravagga of S.N. or mahāvagga of S.N. for student in S.N. faculty, before half of A.N. or after half of A.N. or ekakanipāta+dukanipāta of A.N. for student in A.N. faculty, jātaka+commentary (because kammaṭṭhāna was described in commentary) for student in jātaka faculty.

Qualities of bhikkunovdaka-bhikkhu (teaching bhikkunīs)
If above layman's teachers want to teach bhikkhunī, they must increase their skill level to all of these qualities:

1. Proficient to recite whole tipitaka-pali and commentary-pali. Or at least, he still must recite whole tipitaka, but he can recite just one commentary of suttanta, first 4 parts of commentary of 7 parts of abhidhamma. However, vinaya-commentary is what he must recite it all.

Reference: tipitaka and commentary of vinaya pācittiyakaṇḍa bhikkhunovādakasikkhāpada and vinaya mahāvagga mahākhandhaka.

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Related topic: https://buddhism.stackexchange.com/a/22917/10100