You can't, at the moment. Use Picasa on your local computer.
I loved this online feature too, and it's sad that it's gone. However, seeing that the capability appeared only online, and later in the application, I suspect that they removed it from online because it's difficult to merge your face/name assignments from online into your local Picasa installation. It's a lot easier, and more natural, to do the tagging before uploading.
You can easily do this with Google's command line tools.
http://code.google.com/p/googlecl/
Here's how to create a new album and upload new files to it:
Create an album titled "funny-pics-from-reddit"
stu@sente ~ $ google picasa create --title "funny-pics-from-reddit"
Created album: https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE
Upload some images to that album
stu@sente ~ $ google picasa post --title "funny-pics-from-reddit" *.jpg
Loading file 1IH46.jpg to album funny-pics-from-reddit
Loading file 8518.jpg to album funny-pics-from-reddit
Loading file 8mWVg.jpg to album funny-pics-from-reddit
Loading file aMgoD.jpg to album funny-pics-from-reddit
Loading file B5OzH.jpg to album funny-pics-from-reddit
Loading file BdA1X.jpg to album funny-pics-from-reddit
Loading file cinkh.jpg to album funny-pics-from-reddit
Loading file Il45S.jpg to album funny-pics-from-reddit
Loading file s8mYA.jpg to album funny-pics-from-reddit
Loading file VNXqI.jpg to album funny-pics-from-reddit
Loading file YS0ri.jpg to album funny-pics-from-reddit
Thats all!
you can also get a link for sharing
stu@sente ~ $ google picasa list-albums --title 'funny-pics-from-reddit"
funny-pics-from-reddit,https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE
or get a link for each of images within the album
stu@sente ~ $ google picasa list -link --title "funny-pics-from-reddit"
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197402363448466
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197407826357506
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197411671223314
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197415991037282
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197418880127170
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197428322810098
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197433154709954
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197440054494146
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197445495462290
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197447926728866
https://picasaweb.google.com/108147164794119808099/FunnyPicsFromReddit?authkey=Gv1sRgCNeGh-vqrJLlrgE#5676197458703531074
Best Answer
You don't need to switch to Google+ (although you should probably start considering doing so - it is clear that most new work is going into Google+ Photos and PicasaWeb is on the way out).
You can go directly to https://picasaweb.google.com/?noredirect=1 and it will leave you at the PicasaWeb album page without trying to redirect you to Google+ Photos.