I was disappointed that allmusic.com, my favorite reference source, doesn't even have their full names. But since "Eddie Brown" is given as the author of a couple of songs, maybe we can assume that Eddie's name was Brown?

It lists 6 original albums, all on the Crescendo label:
1964 - There's a Meetin' Here Tonight
1965 - Joe and Eddie Live in Hollywood
1965 - Tear Down the Walls
1965 - Walkin' Down the Line
1966 - The Magic of Their Singing
[no date] Coast to Coast

And one compilation, on the GNP label:
1967 - The Best of Joe & Eddie

Only the "Best of" album has been reissued as a CD (in 1997), and it is the only album for which allmusic provides a track list:

1 There's a Meetin' Here Tonite - Gibson - 2:40
2 Done Laid Around (Gotta Travel On) Clayton, Hayes, Hays, Seeger - 2:55
3 Children Go Where I Send Thee! - - 3:32
4 The Drinking Gourd - Campbell - 3:10
5 Lonesome Traveler - Hays - 2:23
6 The House of the Rising Sun - - 3:07
7 What's That I Hear? (Freedom Calling) - Ochs - 2:18
8 Willie Jean - Eddie Brown - 2:21
9 Mariah - Lerner, Loewe - 2:38
10 Down by the Riverside - - 2:44
11 Pearly Shells - Edwards, Pober, Williams - 1:47
12 Ox Driver's Song - Weaver, Weaver & Lomax - 3:03
13 The First Time Ever I Saw Your Face - MacColl - 2:34
14 I Got Shoes - - 1:50
15 Summertime - Gershwin, Gershwin, Heyward - 3:17
16 Green Grass - - 2:17
17 Sing Hallelujah - Settle - 3:54
18 Roll On - - 2:50
19 Joshua - - 1:58
20 I'm a Drifter - Edmonson - 1:58
21 Michael, Row the Boat Ashore - Joe & Eddie, Baxter, Les Chorus 2:22
22 The Old Man - Kollander - 2:18
23 Work Song - Adderley, Brown - 2:55
24 Tzena, Tzena, Tzena - Issachar, Miron, Mitchell, Parish, Yehiel 2:08
25 The Things I've Saved - Edmonson - 2:49